Fox News has demanded that Tucker Carlson stop posting videos to Twitter, escalating the dispute between the network and its former star host over how — and if — he can continue to speak publicly now that his prime-time show is off the air.

In a letter sent to Mr. Carlson from Fox lawyers, the network accused him of violating the terms of his contract, which runs until early 2025 and limits his ability to appear in media other than Fox. The letter is labeled “not for publication,” in all caps.

Since Mr. Carlson was ousted by Fox News, he has begun producing a bare-bones version of his Fox program, “Tucker Carlson Tonight,” and posting it directly to Twitter. The new show, “Tucker on Twitter,” bears some of the hallmarks of his prime-time show on Fox, including a monologue focused on current affairs and cultural issues.

Harmeet K. Dhillon, a lawyer representing Mr. Carlson, said in a statement that Fox News’s legal threat was not in the interest of the network’s audience.

“Doubling down on the most catastrophic programming decision in the history of the cable news industry, Fox is now demanding that Tucker Carlson be silent until after the 2024 election,” the statement read. “Tucker will not be silenced by anyone.”

Justin Wells, Mr. Carlson’s former executive producer, said on Twitter that the next episode of Mr. Carlson’s show, expected on Tuesday, would feature Mr. Carlson’s response to the indictment of former President Donald J. Trump.

Axios earlier reported that Fox had sent Mr. Carlson the cease-and-desist letter.

A spokeswoman for Fox News said the network had no comment.

Fox’s decision to take Mr. Carlson off the air shocked him and the wider media and political world when the network announced the move in a terse, four-sentence statement. A series of public relations and management headaches led to his downfall, according to interviews with people inside the company. Much to the dismay of senior Fox executives, including the chief executive of Fox Corporation, Lachlan Murdoch, Mr. Carlson continued to push conspiracy theories about the Jan. 6, 2021, assault on the Capitol. A former Carlson producer filed a lawsuit alleging he allowed a hostile, sexist workplace to flourish.

And as part of the defamation lawsuit against Fox by Dominion Voting Systems, which the network settled in April for $787.5 million, Mr. Carlson’s personal text messages became public, revealing how he spoke disparagingly about Fox executives and Mr. Trump.

After the discovery of one particularly incendiary text from Mr. Carlson, the Fox Corporation board decided to begin an internal investigation into his conduct. A few days later, he was out.

Mr. Carlson’s cancellation — he is still an employee of Fox unless the network decides to let him out of his contract — has upended Fox’s lucrative and popular prime-time lineup. Fox, the perennial top-rated cable news network, has seen its lead over its competitors slide. Roughly one-third of its prime-time audience has tuned out since Mr. Carlson was taken off the air.

Fox News had been loath to pick a public fight with its former star, who has commanded one of the biggest followings in conservative media and demonstrated a distinctive ability to shape some of the biggest policy debates in Republican politics. The network has been publicly silent as Mr. Carlson and his associates threatened — mostly through anonymous leaks in the media — to release their own programming and to attack Fox in the process.

Fox’s attempt to force Mr. Carlson off Twitter indicates that talks between his lawyers and representatives for the network are becoming strained and that Fox executives are becoming more pessimistic about reaching an amicable agreement on the terms of his departure.

Mr. Carlson’s lawyers have argued that Fox News breached its contract with Mr. Carlson first, in part by failing to prevent his private messages from being disclosed. The former Fox host also believes his Twitter show is protected speech under the First Amendment, according to a person with knowledge of Mr. Carlson’s legal strategy.

Mr. Carlson’s Twitter videos — recorded at his studio in Maine — have received considerable attention from the news media. Twitter does not release data on how much time its users spend watching videos, making it impossible to know how many people watched Mr. Carlson’s posts for any extended period.

Mr. Carlson joins a long list of television news stars who have tried to use their fame to reach a large audience without the platform of a major broadcast or cable network behind them. Success is not easy to quantify. Katie Couric of CBS News and Ted Koppel of ABC News found it difficult to replicate the clout they once enjoyed after they branched out into their own ventures. A series of former Fox News stars have also seen their influence diminished, including Glenn Beck, Bill O’Reilly and Megyn Kelly.

Even Oprah Winfrey struggled to convert her enormous star power into ratings with the launch of her cable network, OWN.

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