But Mr. Nadler indicated that he was not among the Democrats hungering for an impeachment effort. Attempting to oust Mr. Trump without “overwhelming” evidence of wrongdoing, Mr. Nadler said, would risk “tearing the country apart.”

“You don’t have impeachment unless the case is so strong that you will convince a good fraction — not a majority, necessarily, but a good fraction — of the people who voted for Trump that you had to do it,” Mr. Nadler said.

Senator John Kennedy, Republican of Louisiana, said he expected voters would ultimately render judgment on Mr. Mueller’s findings. Expressing indifference to Mr. Giuliani’s pronouncements, Mr. Kennedy said he hoped Mr. Mueller would give a full report on Russian interference in American elections, and soon.

“Once he finishes the investigation, I want him to report to the American people and let the chips fall where they may,” Mr. Kennedy said of the special counsel. “And I trust the American people to figure it out.”

But Mr. Kennedy does not face re-election until 2022, and few politicians on the ballot this fall share his serene outlook. While some candidates on the left and right have sought to exploit the Mueller investigation in different ways — Democrats calling for impeachment, Republicans demonizing the special counsel — most politicians in difficult races have sought to avoid the issue. They have largely adopted a posture of deference to Mr. Mueller, insisting that the special counsel must finish his work before they judge the facts.

That day of judgment, however, may be approaching with inconvenient speed. Should Mr. Mueller unearth information implicating the president or members of his immediate family in serious crimes, it could put enormous and in many cases unwanted pressure on Congress to take action — and on congressional candidates to take a stand.

Mr. Trump and his lawyers have denied that he conspired with Russians to influence the 2016 election, or that he did anything to obstruct an investigation into Russian interference.