“Republicans who are reflexively defending the self-inflicted wounds of this president have no need for him with Mike Pence in the wings,” Erick Erickson wrote on Wednesday on his conservative blog, The Resurgent.

Mr. Pence takes umbrage at the very mention of the idea, according to people who have heard him speak about it. And Republican lawmakers are eager to tamp down on any discussion of it, particularly after Mr. Rosenstein on Wednesday named a special counsel to lead the investigation into whether Mr. Trump’s campaign colluded with Moscow to influence the election.

“I’m not even going to give credence to that,” the House speaker, Paul D. Ryan of Wisconsin, said on Thursday, when asked by a reporter about colleagues who are privately contemplating a Pence presidency. “I’m not even — there’s not even a point making a comment on that.”

Still, the vice president may have fed speculation when he filed paperwork this week for a political action committee, an unusual move for a vice president.

Through it all, Mr. Pence has played the part of loyal soldier and messenger, gamely going about a workaday schedule that includes delivering Mr. Trump’s greetings to activists, lobbyists and lawmakers. He projects a “this is fine” outlook on the administration’s agenda of overhauling health care and the tax system.

“Whatever Washington, D.C., may be focused on at any given time, rest assured, President Donald Trump will never stop fighting for the issues that matter most to the American people — good jobs, safe streets and a boundless American future,” Mr. Pence told executives and lobbyists on Thursday at an investment summit meeting at the U.S. Chamber of Commerce.

A block away at the White House, Mr. Trump took a far different tack, telling a group of news anchors that the appointment of a special counsel to investigate his campaign’s possible collusion with Russia “hurts our country terribly.” Not long after that, with Mr. Pence in the front row at an East Room news conference, Mr. Trump declared that the inquiry was “a witch hunt” and that there was “no collusion” between him and Russia — though he quickly added, “I can always speak for myself.”