Rudolph W. Giuliani, who served as former President Donald J. Trump’s personal lawyer, was interviewed last week by federal prosecutors investigating Mr. Trump’s efforts to overturn the 2020 election, people familiar with the matter said.

The voluntary interview, which took place under what is known as a proffer agreement, was a significant development in the election interference investigation led by Jack Smith, the special counsel, and the latest indication that Mr. Smith and his team are actively seeking witnesses who might cooperate in the case.

The session with Mr. Giuliani, the people familiar with it said, touched on some of the most important aspects of the special counsel’s inquiry into the ways that Mr. Trump sought to maintain his grip on power after losing the election to Joseph R. Biden Jr.

“The appearance was entirely voluntary and conducted in a professional manner,” said Ted Goodman, a political adviser to Mr. Giuliani.

A proffer agreement is an understanding between prosecutors and people who are subjects of criminal investigations that can precede a formal cooperation deal. The subjects agree to provide useful information to the government, sometimes to tell their side of events, to stave off potential charges or to avoid testifying under subpoena before a grand jury. In exchange, prosecutors agree not to use those statements against them in future criminal proceedings unless it is determined they were lying.

Prosecutors working for Mr. Smith asked Mr. Giuliani about a plan to create fake slates of pro-Trump electors in key swing states that were actually won by Mr. Biden, one person familiar with the matter said, speaking on the condition of anonymity to discuss an ongoing criminal investigation. They focused specifically on the role played in that effort by John Eastman, another lawyer who advised Mr. Trump about ways to stay in office after his defeat.

Mr. Giuliani also discussed Sidney Powell, a lawyer who was briefly tied to Mr. Trump’s campaign and who made baseless claims about a cabal of foreign actors hacking into voting machines to steal the election from Mr. Trump, the person said.

Ms. Powell, who was sanctioned by a federal judge for promoting conspiracy theories about the voting machines, also took part in a meeting in the Oval Office in December 2020 during which Mr. Trump was presented with a brazen plan — opposed by Mr. Giuliani — to use the military to seize control of voting machines and rerun the election.

The person said that prosecutors further asked Mr. Giuliani about the scene at the Willard Hotel days before the attack on the Capitol. Mr. Giuliani and a group of close Trump advisers — among them, Mr. Eastman, Mr. Trump’s former chief strategist Stephen K. Bannon and Mr. Trump’s current adviser Boris Epshteyn — had gathered at the hotel, near the White House, to discuss strategies before a violent mob stormed the Capitol on Jan. 6, 2021, disrupting the certification of Mr. Biden’s victory over Mr. Trump.

The proffer session with Mr. Giuliani, elements of which were reported earlier by CNN, came as Mr. Smith’s team pressed ahead with its election interference inquiry of Mr. Trump even as it prepares for the former president’s trial on separate charges of putting national security secrets at risk and obstructing government efforts to recover classified documents.

The prosecutors have been bringing witnesses before a grand jury and conducting separate interviews of others as they seek to assemble a fuller picture of the various ways in which Mr. Trump and his allies were promoting baseless claims that the election had been stolen from him and seeking to reverse his electoral defeat.

In some cases, they appear to be gauging whether they can elicit useful information without necessarily agreeing to formal cooperation deals.

Last week, The New York Times reported that prosecutors were in negotiations to reach a proffer agreement with Michael Roman, the former director of Election Day operations for Mr. Trump’s 2020 campaign. Mr. Roman was also instrumental in helping put together the so-called fake elector plan.

The push to assemble slates of pro-Trump electors from swing states won by Mr. Biden is one of a number of components of Mr. Smith’s investigation. Prosecutors have also scrutinized whether Mr. Trump and his allies bilked donors by raising money through false claims of election fraud, examined efforts to use the Justice Department to give credence to election-fraud claims and sought to piece together a detailed picture of the role played by Mr. Trump in inciting the attack on the Capitol and the disruption of the congressional certification of his loss.

It remains unclear whether Mr. Giuliani will face charges in the special counsel’s investigation. He is also under scrutiny on many of the same subjects by the district attorney in Fulton County, Ga., who is pursuing a wide-ranging investigation into Mr. Trump’s effort to reverse his election loss in that swing state.

As part of Mr. Smith’s inquiry, prosecutors questioned Mr. Roman’s deputy, Gary Michael Brown, last week in front of a grand jury in Federal District Court in Washington that has been investigating the attempts by Mr. Trump and his allies to overturn the election. Federal prosecutors on Wednesday are also scheduled to interview Brad Raffensperger, the secretary of state of Georgia, who took a call from Mr. Trump in early January 2021 during which the former president asked him to “find” sufficient votes that would put him over the top in the election in that state.

A longtime ally of Mr. Trump who served two terms as New York City’s mayor, Mr. Giuliani effectively led the former president’s attempts to overturn his defeat in the last presidential race and has for months been a chief focus of the Justice Department’s broad investigation into the postelection period. His name has appeared on several subpoenas sent to former aides to Mr. Trump and to a host of Republican state officials involved in the plan to create fake slates of electors.

Last year, shortly before Mr. Smith was appointed to his job as special counsel, the Justice Department issued a subpoena to Mr. Giuliani for records related to his representation of Mr. Trump, including those that detailed any payments he had received. A group of federal prosecutors including Thomas Windom had been pursuing various strands of the inquiry into Mr. Trump’s efforts to remain in power before Mr. Smith’s appointment and they continue to play key roles in the investigation.

Among the things that prosecutors have been examining are the inner workings of Mr. Trump’s fund-raising vehicle, Save America PAC. The records subpoenaed from Mr. Giuliani could include some related to payments made by the PAC, according to a person familiar with the matter.

More recently, prosecutors have been asking questions about Mr. Trump’s false claims that his defeat in the election was caused by widespread fraud, and how he aggressively raised money off those claims. The prosecutors have drilled down on the issue of whether people around Mr. Trump knew that he had lost the race, but continued raising money off the fraud claims anyway.

The House select committee that investigated the attack on the Capitol on Jan. 6 first raised questions publicly about Mr. Trump’s fund-raising, and the special counsel’s team has picked up on that thread. Among other questions they have asked witnesses is whether their lawyers are being paid for by the political action committee that became a repository for money raised off Mr. Trump’s false claims of widespread fraud.

Investigators have walked through a timeline with various witnesses, including asking people about election night and what Mr. Giuliani may have been telling Mr. Trump before his defiant speech declaring he had won the election, as well as about Jan. 6 and Mr. Trump’s actions that day.

The special counsel’s office has focused on Mr. Trump’s mind-set and who was telling him he lost, according to people familiar with the questions. Among the questions has been whether there were concerns raised among people working with the campaign as to the language used in television ads about fraud in December 2020, and who signed off on the ad copy.

Prosecutors also subpoenaed former Vice President Mike Pence, who was a key focus of Mr. Trump’s efforts to stay in power as Mr. Trump tried to pressure him to use his ceremonial role overseeing congressional certification to block Mr. Biden from being certified.

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