Many of the arguments and much the day was tailored to convincing a few Republicans who remained holdouts on the question of whether to call witnesses. Mr. McConnell gave his party’s first question on Wednesday to Senators Mitt Romney of Utah, Susan Collins of Maine and Lisa Murkowski of Alaska in an effort to allay the concerns of the three lawmakers, Republican moderates who are swing votes on the issue. The trio teamed up to ask Mr. Trump’s lawyers how they should judge the president if they conclude he acted in the Ukraine matter with both political and policy motives.

The selection of Ms. Collins, who is facing the toughest re-election campaign in her long Senate career, was revealing: It suggested that Mr. McConnell was keenly focused on giving her every opportunity to have her voice heard before moving forward. When she rose to announce herself, she became the first senator other than the two leaders to have a speaking role in the trial.

Notably absent from the group was the fourth Republican who had expressed interest in witnesses: Senator Lamar Alexander of Tennessee, a close friend of Mr. McConnell’s who has said he will not decide whether to support witnesses until after the question period has closed.

Later in the evening, Mr. Romney hinted that he was seeking to better understand why Mr. Trump withheld security aid for Ukraine, asking White House lawyers for the specific date that the president first ordered a hold on the assistance. Patrick Philbin, a deputy White House counsel, did not provide a direct answer, saying there was no evidence of a specific date and offering only testimony showing that budget officials were aware of the hold in late June.

On Tuesday, Mr. McConnell had privately warned his rank-and-file members that he did not currently have the votes to stop Democrats from summoning witnesses. One after the other on Wednesday, in statements and interviews in the Capitol, they made clear they would side with their leader.

Senator Patrick J. Toomey of Pennsylvania, who had previously floated the idea of a witness deal, said Wednesday that he was “very, very skeptical” of new witnesses. Senator Cory Gardner of Colorado, facing a tough re-election in a swing state, issued a statement saying that he had heard enough and would vote against hearing from anyone else. Senator Martha McSally of Arizona, who is also facing a difficult election, tweeted later Wednesday: “I have heard enough. It is time to vote.”

Mr. McConnell summoned Ms. Murkowski to his office on Wednesday morning for a private meeting before the trial began, and she emerged refusing to answer questions about whether she would ultimately support the call for witnesses.