Democrats in the Senate want to call Mr. Bolton to the stand, compel the testimony of three other top Trump aides, including the acting chief of staff, Mick Mulvaney, and obtain the records that the administration has withheld. But they would need the support of at least four Republican senators to do so.

“A fair trial, everyone understands, involves evidence,” Senator Richard J. Durbin, Democrat of Illinois, said Sunday on NBC’s “Meet the Press.” “Evidence would be documents and witnesses. We know the president has refused to provide documentation beyond the July 25 telephone memo. And he’s refused to provide basic witnesses who actually heard what happened on that conversation and saw what happened afterwards.”

Here are some of the key questions that more witness testimony or additional documents could address:

To what extent did Mr. Giuliani act at Mr. Trump’s direction and keep him apprised of his efforts?

Some Trump allies have tried to suggest that Mr. Giuliani was a rogue actor pursuing his own interests in Ukraine. But Mr. Giuliani said last spring, as he planned a trip to Ukraine to press for investigations into the Bidens, that his efforts had Mr. Trump’s full support and that the president “basically knows what I’m doing.”

Mr. Trump has called Mr. Giuliani a “crime fighter” who was “seeking out corruption” because he was “very, very incensed at the horrible things that he saw.” He also said Mr. Giuliani had the right to look into whether the Ukrainians helped sow the seeds for the special counsel’s investigation into Russia’s interference in the 2016 election.

Mr. Giuliani has insisted that his conversations with Mr. Trump are protected by attorney-client privilege. He has pointed out that Kurt D. Volker, then Mr. Trump’s special envoy to Ukraine, put him in touch with a top Ukrainian aide whom he met in August in Madrid.

Unquestionably, Mr. Trump sought to vest Mr. Giuliani with at least some informal authority to operate on his behalf. He urged President Volodymyr Zelensky of Ukraine, during a phone call on July 25, to consult Mr. Giuliani about the investigations he wanted. And he urged his own diplomats and aides involved with Ukraine to consult with Mr. Giuliani after they returned from Mr. Zelensky’s inauguration in May.