Democrats have been pushing for four witnesses — including John R. Bolton, Mr. Trump’s former national security adviser, who wrote in an unpublished manuscript that the president conditioned security aid to Ukraine on investigations into Democrats — over the strong objections of Senator Mitch McConnell, the majority leader. Some Republicans are floating the idea of a witness swap in which they would call either former Vice President Joseph R. Biden Jr. or his son Hunter Biden, both of whom Mr. Trump wanted Ukraine to investigate, even though neither has direct knowledge of Mr. Trump’s behavior.

Democrats have opposed such a move, and Mr. Schiff suggested on Sunday that Chief Justice John G. Roberts Jr., who is presiding over the trial, should rule on that question.

“We have a very capable justice sitting right behind me who can make decisions about the materiality of witnesses,” Mr. Schiff said, adding, “We trust the Supreme Court justice.”

If history is any guide, Chief Justice Roberts will be reluctant to do so. When President Bill Clinton was tried in the Senate in 1999, Chief Justice William H. Rehnquist used his authority sparingly, leaving most questions to the Senate to decide.

Lawmakers on both sides — along with Alan Dershowitz, a consultant to Mr. Trump’s legal team — took to the Sunday morning talk show circuit to make the case for or against Mr. Trump. The president was impeached by the House in December on charges that he abused his oath of office and obstructed Congress by pressuring the leader of Ukraine to investigate his political rivals and then covering it up by concealing evidence from lawmakers.

Mr. Schiff and his team of prosecutors maintain that the president was trying to influence the 2020 election for his personal gain. During an abbreviated session of the Senate on Saturday, the president’s team pushed back hard on that assertion, arguing that it was the Democrats who were trying to undo the results of the 2016 election — and to interfere with the one in 2020.

“They’re asking you to tear up all of the ballots all across the country on your own initiative, take that decision away from the American people,” Pat A. Cipollone, the White House counsel, said of the House managers, adding: “They’re here to perpetrate the most massive interference in an election in American history, and we can’t allow that to happen.”