Over the summer, Mr. Trump publicly dangled the possibility of firing Mr. Mueller, stoking concerns among some of his advisers who believed that Mr. Trump would further imperil his presidency if he did so.

But two people who have spoken to the president recently said that he was far more frustrated with the attorney general, Jeff Sessions, and the F.B.I. director, Christopher A. Wray, than Mr. Mueller. Mr. Trump has said that Mr. Wray has not moved quickly enough to rid the bureau of senior officials who were biased against Mr. Trump and had worked for James B. Comey, the director whom Mr. Trump fired in May.

Mr. Trump’s lawyers assured the president throughout the fall that Mr. Mueller’s investigation would be over by the end of the year. But on Dec. 1, Mr. Flynn entered his guilty plea and agreed to cooperate with Mr. Mueller’s investigation, an indication that the inquiry will not soon close.