“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.