Senator Rick Scott, Republican of Florida, is proposing a constitutional amendment to raise the voting threshold for passing articles of impeachment in the House to three-fifths, from a simple majority, because of what he saw as a partisan attempt to impugn Mr. Trump.

On the Democratic side, Senator Jeff Merkley of Oregon wants to institute a rule that would guarantee new witnesses and documents to be allowed, after Republicans led by Senator Mitch McConnell of Kentucky, the majority leader, thwarted Democratic attempts to do so. Senator Chuck Schumer of New York, the Democratic leader, endorsed the concept.

“I’d like to see witnesses and documents be required,” Mr. Schumer said.

Senator Josh Hawley, Republican of Missouri, wants to “deem” any House-passed articles of impeachment as sent to the Senate within a certain time period even if the House does not formally send them — a response to Speaker Nancy Pelosi’s decision to sit on the House-passed accusations against Mr. Trump for weeks. Republicans saw that as an unacceptable attempt by the House to dictate to the Senate the trial’s timing and ground rules.

“There ought to be some way to get the trial started, you would think, short of the physical papers being delivered,” Mr. McConnell, who supports Mr. Hawley’s idea, said in an interview.

Despite dissatisfaction in both parties about how parts of Mr. Trump’s trial unfolded, some senior lawmakers warn that the Senate should not act hastily in making underlying changes to the rules when emotions remain raw. One lawmaker’s nuisance impeachment would no doubt be another’s necessary one.