Several of Mr. Trump’s former aides have struck deals with the special counsel team in exchange for their cooperation in the investigation. This week, Jerome Corsi, a friend of Roger J. Stone Jr., a longtime informal adviser to the president, announced that he expected to be charged with lying to federal investigators after two months of cooperation with the special counsel team.

Mr. Mueller, whose appointment did not require approval from the Senate, has sought to ask the president about any possible coordination with Russia during the campaign and whether he intended to obstruct the investigation.

The special counsel team has negotiated over a possible presidential interview for nearly a year. After months of back-and-forth with the president’s personal lawyers, Mr. Mueller’s team agreed to accept answers in writing about possible conspiracy with Russia. On obstruction matters, investigators acknowledged that issues of executive privilege could complicate the prospect of an interview, but they did not abandon pursuit of it.

The president’s lawyers are said to believe that the special counsel investigation is winding down, though Mr. Mueller’s team has given no public indication about its timeline.

Mr. Trump’s lawyers have blanched at the idea of the president sitting down and being interviewed by Mr. Mueller and his team out of concern that the president might make a false statement. Lying to federal investigators is a felony that could carry up to five years in prison.

Mr. Trump’s comments on Thursday renewed concerns on Capitol Hill that the president could move to fire Mr. Mueller and end his investigation.

Democrats and a handful of Republicans are pushing for Congress to vote before the end of their session next month on legislation providing job protections to special counsels like Mr. Mueller.