Shortly after his plane took off earlier this month from Riyadh, where he had held a lengthy meeting with Saudi Arabia’s crown prince, Secretary of State Antony J. Blinken called a different Middle East leader, Prime Minister Benjamin Netanyahu of Israel.

Over 40 minutes, Mr. Blinken gave the Israeli leader a briefing about the significant demands the young crown prince, Mohammed bin Salman, was making for his nation to normalize diplomatic relations with Israel. Mr. Netanyahu had an update on his own demands.

The phone call — described by two American officials — was a turn in the Biden administration’s long-shot bid to broker a landmark diplomatic deal between Saudi Arabia and Israel, two historical adversaries who in recent years have been engaged in a discreet courtship in part over their shared distrust of archrival Iran.

The White House, which for more than two years has largely been content to sit out the poker game of Middle East diplomacy, has decided to make a bet and push some of its chips in. The United States is now in the midst of complex negotiations among three leaders who have their own reasons for a deal but are making demands that might prove to be too costly. And they simply do not much like or trust each other.

Several senior American officials said the chances of a deal could be less than 50 percent, and Mr. Blinken said he had “no illusions” the path to a deal would be quick. Still, a normalization of relations between Saudi Arabia and Israel would be one of the most dramatic events in a continued realignment of the Middle East, and could reap benefits for leaders of both countries, as well as President Biden, who faces re-election next year.

It would also make explicit what has been true for a long time: that the government of one of the Arab world’s most influential countries has effectively made its support for a Palestinian independent state a lower priority.

For Mr. Netanyahu, Saudi Arabia’s recognition of Israel would be a significant political victory for the embattled leader, whose hard-right coalition government faces fierce domestic opposition.

For his part, Prince Mohammed is seeking a strengthened security relationship with the United States, access to more American weapons and U.S. consent for the kingdom to enrich uranium as part of a civilian nuclear program — something that Washington has long resisted.

For Mr. Biden, drawing closer to Saudi Arabia carries political risks — he once pledged to make Saudi Arabia a “pariah” — but a diplomatic pact in the Middle East could be a boon ahead of the 2024 election. American officials also see strategic importance in bolstering ties with Saudi Arabia: as a way to keep Riyadh from gravitating further toward China, two nations that have engaged in an increasingly warm embrace.

Working against the prospects of a deal is the fact that all sides would have to reverse course on at least one long-held position: for Israel, that the country would never allow nuclear enrichment in the Saudi kingdom; for Saudi Arabia, that peace with Israel can only come after an established state for the Palestinian people; for Mr. Biden, cementing a closer alliance with Saudi Arabia would force him to make a public case for why he changed his position on Prince Mohammed..

Several American officials described the current push by the Biden administration, and the chances for success, on the condition of anonymity because of the sensitivity of the negotiations. They said they believe a confluence of factors have created a window of time — perhaps before next year when the American election cycle intensifies — to pull together a possible accord. One of the factors is that a Democratic president might have a better chance than a Republican president of selling the deal to party members and bringing some in the political opposition along.

In recent weeks the Biden administration has accelerated the cadence of top officials traveling to Riyadh and Tel Aviv to meet with Prince Mohammed and Mr. Netanyahu. This week, just days after Mr. Blinken’s visit, Brett McGurk, the top White House official handling Middle East policy, led a delegation on an unpublicized trip to continue the negotiations, according to two American officials. Jake Sullivan, the national security adviser, traveled to Saudi Arabia in May.

“Biden has decided to go for it, and everyone in the administration now understands that the president wants this,” said Martin Indyk, a former U.S. ambassador to Israel, who adds that a committed American president has long been essential for diplomatic breakthroughs between Israel and Arab nations. “When you’re talking about Middle East peace, it takes three to tango.”

A new defense pact or nuclear deal with Saudi Arabia would face another hurdle: getting approval from a sharply divided Congress in which some prominent members of Mr. Biden’s party would likely vote against it. But odd political alliances have also formed, with one prominent Republican senator, Lindsey Graham of South Carolina, quietly assisting the White House’s negotiations.

The Saudi embassy in Washington did not respond to a request for comment. A representative for the National Security Council said that the Biden administration’s Middle East policy “includes efforts to expand and strengthen the Abraham Accords,” as well as efforts to normalize relations between Israel and Saudi Arabia. Mr. Netanyahu has made no secret of his hope to seal a deal with the Saudis.

The prospect of a formal rapprochement between Israel and Saudi Arabia has percolated for years, but both sides have seen too many obstacles to make the idea a reality. When President Trump in September 2020 presided over the signing of the Abraham Accords — diplomatic agreements between Israel and two Gulf Arab nations — the Saudis were not ready to join the pact.

Though Mr. Biden took a frosty attitude toward Prince Mohammed, known as MBS, in part over the killing of the journalist and Washington Post contributor Jamal Khashoggi, he reluctantly visited the kingdom last July. Relations between the two countries hit a nadir in October, when the Saudis announced they were cutting oil production, a move that blindsided American officials.

The two governments made quiet efforts to repair relations over the winter. Then in May, when Mr. Sullivan, the national security adviser, visited Riyadh, Prince Mohammed indicated a greater willingness to normalize relations with Israel. He agreed with Mr. Sullivan that this year might be the time to do it — but for the right price, said two people familiar with what transpired on the trip. This message, which Mr. Sullivan conveyed to Mr. Biden, seems to have swayed the president to make a push on a deal.

This led to the visits to Riyadh this month of Mr. Blinken and Mr. McGurk.

For Saudi Arabia, normalization with Israel is less about Israel and more about what it can get out of the United States, its historical security guarantor. Given how unpopular Israel remains among Saudi citizens, normalizing relations with the country would cost Prince Mohammed political capital with his own people, Saudi officials say. To justify that, they say, he would need to secure significant concessions from the United States, with an eye toward deterring Iran.

But Prince Mohammed’s initial demands were steep: U.S. guarantees to defend Saudi Arabia from military attack, a Saudi-American partnership to enrich uranium for a civilian nuclear program and fewer restrictions on U.S. arms sales to the kingdom.

Richard Goldberg, a White House official during the Trump administration and now a senior adviser at the Foundation for Defense of Democracies, which advocates for greater security for Israel, met with senior Saudi officials last month. In an interview, he said officials talk about uranium mining and enrichment for export revenue, but he believes that glosses over the real purpose: to have the means to build up a nuclear arsenal if Iran does the same.

“The open question — the big question mark — is this: Is the uranium enrichment a red line, as MBS says, or is it an opening position?” said Mr. Goldberg, who is opposed to Iranian enrichment and has “strong discomfort” over the prospect of Saudi enrichment. “Whether it’s a bargaining position or truly a red line is not really known.”

Mr. Netanyahu is in the most serious political crisis of his years as prime minister: he is on trial for corruption and the legal reform he tried to pass was met with mass protests in the country. If new elections were held tomorrow, polls suggest Mr. Netanyahu would lose.

A landmark diplomatic arrangement could help reverse his political fortunes, some close to him believe. But consenting to a Saudi nuclear enrichment program would also be a reversal of longstanding policy in Israel, which worries that a Saudi nuclear program could lead to a nuclear arms race across the Middle East.

Mr. Netanyahu could still face strong opposition from Israel’s national security establishment if he agrees to a Saudi enrichment program.

A small group of Israeli aides has been entrusted to handle negotiations over a possible Saudi deal, including Ron Dermer, the minister of strategic affairs and a former ambassador to Washington, and Tzachi Hanegbi, the national security adviser. The group has visited Washington several times in recent months.

With a hard-line government in Israel, there are no prospects for any deal that makes provisions for a Palestinian state. But for a rapprochement between Saudi Arabia and Israel to take place, the Saudis and the Biden administration have insisted that any deal includes some concrete gestures for the Palestinians, officials say.

What those might be remains unclear.

“Bibi wants this so badly he can taste it,” said Mr. Indyk. But, he said, unless there were real accommodations made by Israel toward the Palestinians, the deal would be ephemeral and U.S. concessions to the Saudis would be wasted. “The Saudis are supposed to deliver the Muslim world, but if the U.S. lets MBS leave the Palestinians behind, the whole thing becomes unstable.”

In public, Saudi officials have repeatedly said that they will not establish relations with Israel without a deal that includes the creation of a Palestinian state — a line they have maintained since the kingdom led the 2002 Arab peace initiative, which offered Israel diplomatic relations with Arab countries in exchange for the establishment of a Palestinian state. Prince Mohammed reiterated that message at an Arab League summit last month.

“The Palestinian cause was, and still is, the pivotal issue for Arabs and Muslims,” Prince Mohammed said. “It comes at the top of the kingdom’s foreign policy priorities.”

Among the largest barriers to Saudi Arabia expanding its ties with Israel is public opinion. Even as the Gulf’s authoritarian rulers and business elites lean toward deepening their relationships with Israel, most Gulf citizens are opposed to full normalization. In an April poll by the Washington Institute, 78 percent of Saudis said the Abraham Accords would have a negative impact on the region.

Any new defense pact or nuclear deal with the Saudis would require congressional approval, a tall order given the ambivalent or outright hostile attitude of some prominent Democratic lawmakers toward the kingdom.

Along with Republican colleagues, lawmakers have denounced Prince Mohammed for the murder of Mr. Khashoggi — in which he has vehemently denied playing any role — and the mass killing of civilians in the war in Yemen. A top Democrat in the Senate, Robert Menendez of New Jersey, has placed a hold on the sale of certain weapons to Saudi Arabia.

Lawmakers have also expressed their concerns over any move by Saudi Arabia to enrich uranium in its territory, citing proliferation concerns. For years, the State Department has been trying to negotiate what it calls a 123 agreement with the Saudis, which would lay out tough nonproliferation criteria to allow for American cooperation on civilian nuclear energy, though Saudi officials have balked at the restrictions in part because of Iran’s program. The United States has such an agreement with the United Arab Emirates, Saudi Arabia’s neighbor, that bans in-country uranium enrichment.

But Democratic and Republican lawmakers are generally supportive of promoting normalization between Israel and Arab nations, and they know that such accords can be a political gain to win over pro-Israel voters during election seasons.

Aides working for the two top senators on the Senate Foreign Relations Committee, Mr. Menendez and Jim Risch, Republican of Idaho, are drafting a bill that calls for the U.S. government to try to deepen the Abraham Accords and expand what they call “regional integration.”

Among the Republicans, an unlikely figure has stepped forward offering to help: Mr. Graham. In an interview, he said he has been working with top Biden administration officials to help to broker Saudi-Israeli peace.

“Ending the Arab-Israeli conflict would be a game changer for the world and further isolate Iran,” he said.

Mr. Graham says he has spoken to Senator Mitch McConnell, the Kentucky Republican and minority leader, and other top G.OP. lawmakers, and said there would potentially be “a lot of support on the Republican side.” He met with Prince Mohammed in Riyadh earlier this year, and has frequent discussions with senior Israeli officials.

Although he was one of the most strident critics of Prince Mohammed after Mr. Khashoggi’s killing and once called the crown prince “a wrecking ball to the region jeopardizing our national security interests on multiple fronts,” Mr. Graham has now changed his tune.

While “the Khashoggi thing is no small matter,” he said he made a decision to re-engage with Saudi Arabia because it is in the interests of the United States — isolating Iran and possibly blunting China’s influence over Saudi Arabia.

Mr. Graham also said it would also bring credit to former President Trump and Jared Kushner, the former president’s son-in-law, who brokered the diplomatic pacts between Israel and several Arab countries during the final months of his presidency.

He also has parochial business interests: More arms deals with Saudi Arabia could bring economic benefits to his home state. In May, the senator praised Saudi Arabia’s decision to purchase more than $35 billion worth of Boeing Dreamliner jets, which are manufactured in South Carolina.

During a celebratory event at Boeing’s South Carolina plant, he was ecstatic.

“Let it be said that the journey to the future of the Middle East ran through Charleston, South Carolina!” he said.

Eric Schmitt contributed reporting from Washington, and Patrick Kingsley from Jerusalem.

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