On Saturday evening, Mr. Trump for the first time posted a tweet that specifically mentioned the special counsel, Robert S. Mueller III. The president again denounced the inquiry he is conducting as a “witch hunt.”

“The Mueller probe should never have been started in that there was no collusion and there was no crime,” he wrote. “It was based on fraudulent activities and a Fake Dossier paid for by Crooked Hillary and the DNC, and improperly used in FISA COURT for surveillance of my campaign.”

Mr. Dowd, by contrast, did not name the special counsel in his statement, but the implication that he believed Mr. Mueller should be fired was unmistakable. Such a move could set off alarms among Republicans in Congress, who have largely stood by as the president repeatedly assailed the Justice Department and the F.B.I.

Mr. Dowd said at first that he was speaking on behalf of the president but later backed off that assertion. He did not elaborate on why he was calling for the end of the investigation, saying only: “Just end it on the merits in light of recent revelations.”

People close to the president were skeptical that Mr. Dowd was acting on his own. Mr. Trump has a history of using advisers to publicly test a message, giving him some distance from it. And Mr. Dowd’s comments came at a time when members of Mr. Trump’s legal team are jockeying to stay in his favor.