Yet the timing of such an effort, with just 11 days until Mr. Trump is to leave office, scrambled the equation. Senator Mitch McConnell of Kentucky, the Republican leader, indicated that under Senate rules a trial could not begin until Jan. 19, the day before President-elect Joseph R. Biden Jr.’s inauguration, meaning the process would not advance quickly enough to avert any feared dangerous moves in Mr. Trump’s last days in power.

That raised the prospect of conducting a trial after Mr. Trump vacates the White House, overshadowing the opening days of Mr. Biden’s administration at a time when he would like to turn the page and confront crises like the coronavirus pandemic, which has grown even deadlier while attention has focused on Washington’s political wars. A nationally televised trial could dominate discussion and would prevent other business in the Senate.

“If the House does send articles of impeachment over, they really get the Biden administration off to a bad start,” Senator Roy Blunt, Republican of Missouri, said in an interview on Saturday. “Whether that’s the first 10 days or the first 20 days of the Biden administration, it’s certainly not how you’d want to start your presidency off.”

Some of Mr. Trump’s critics argued that it would be important to hold a trial even if he is already out of power in order to bar him from ever seeking office again, a penalty envisioned by the Constitution — and perhaps more important, to render a verdict condemning his actions for the sake of history.