WASHINGTON — The White House and the Republicans in the Senate, all but certain that the House will move forward to impeach President Trump, are divided over whether to embrace a lengthy trial that could give his allies a chance to mount an elaborate defense of his conduct before a polarized nation, or to move quickly to dispense with charges against him.

Several Republican senators discussed the issue with some of Mr. Trump’s top aides on Thursday during a meeting at the White House that unfolded as the House Intelligence Committee capped off two weeks of public impeachment hearings exploring whether the president should be impeached on charges of pressuring Ukraine to announce investigations into his political rivals.

The group, which included some of Mr. Trump’s closest allies in the Senate and his legal and political advisers, came to no final conclusions about what a person briefed on the matter said would be a “totally unpredictable” situation as all 100 senators meet in public session for only the third time in history to consider whether to remove a president from office.

One White House official said nothing would be resolved until closer to the time of the actual trial. Another person familiar with the White House position said that they believe there should not be a vote in the House and that they considered the inquiry illegitimate, but that they welcomed the chance to present witnesses and try the case, which they cannot do in the current setting.

Mr. Trump has told friends that he is eager to see Senate Republicans aggressively argue that he did nothing wrong, after an elaborate House impeachment proceeding that has featured a constant barrage of damaging Democratic allegations.

But some lawmakers, including Senator Lindsey Graham, Republican of South Carolina, are pushing for a quick trial — perhaps as short as two weeks, according to people familiar with the meeting. They hope that a brief proceeding would limit the political damage to Mr. Trump and quickly lead to his acquittal, allowing him and the Republican Party to focus on winning the 2020 election.

Others believe that drawing a trial out for as long as a month early next year could bring political advantages to Republicans, especially if it forces several Democratic senators who are running for president to make a difficult choice between sitting in their seats during an impeachment trial or spending time with voters on the campaign trail in Iowa and New Hampshire.

On Thursday, Mr. Graham sent a letter to Secretary of State Mike Pompeo requesting documents that could signal which witnesses Republicans might call during an impeachment trial and how his allies might seek to defend the president.

In the letter, Mr. Graham asked for documents and communications with former Vice President Joseph R. Biden Jr., his son Hunter Biden, officials from the Obama administration and former President Petro O. Poroshenko of Ukraine. The list suggests that Mr. Graham envisions a defense of Mr. Trump that focuses on shifting attention away from Mr. Trump’s conduct and onto the issue of whether Hunter Biden’s work on the board of a Ukrainian energy company while his father was vice president was appropriate.

House Republicans unsuccessfully tried to convince the Democrats leading the impeachment inquiry to subpoena Hunter Biden.

A longer trial might help a handful of moderate Republican senators who are eager to show independent voters that they are taking the allegations against Mr. Trump seriously. For those senators, a quick dismissal could be seen as a decision to condone the president’s actions and sweep them under the rug without due consideration. Republican leaders have told Mr. Trump that they do not believe they could get the 51 votes required to quickly dismiss potential articles of impeachment.

Mr. Trump himself is a wild card, according to several people familiar with his thinking about how to handle a trial that appears all but inevitable.

At some moments, the president has told people close to him that he wants to see a lengthy trial in which his defenders are given the opportunity to call witnesses and deliver speeches on his behalf. But at other times, as he watches a torrent of negative news on cable television, he has said he wants a quick end to a process that he finds intolerable.

On Thursday morning, he complained on Twitter about coverage of what he called the “phony Impeachment Hoax,” accusing the news media of failing to fairly report about the impeachment hearings. “FAKE & CORRUPT NEWS!”

In a separate lunch meeting with a group of eight senators on Thursday, including some who have signaled an openness to the impeachment inquiry, Mr. Trump made brief comments about his frustration with the process, but did not ask for any commitments and did not discuss the procedures that a trial might follow, according to a statement from Senator Susan Collins, Republican of Maine.

“He feels that whatever has come forward has been exactly what he says — useless,” Senator Shelley Moore Capito, Republican of West Virginia, told reporters after returning from the lunch.

Senator Mitt Romney, Republican of Utah, who has been one of the few members of the president’s party to criticize Mr. Trump’s conduct on Ukraine, attended the lunch but said Mr. Trump shook his hand and did not mention the comments about his conduct.

The staff-level discussions with the senators on Thursday focused on the variety of ways that a Senate trial could play out and decisions that Senator Mitch McConnell, Republican of Kentucky and the majority leader, will eventually have to make about the rules that will govern it.

Those rules, which were last developed for the impeachment trial of former President Bill Clinton in early 1999, govern how the two sides present evidence, whether they can call witnesses, and what role senators would have as they serve as quasi-jurors on the Senate floor. The answers to those questions — which would have to be negotiated with Democratic senators — would affect how long a trial would last.

In Mr. Clinton’s case, senators agreed that Republicans could only call three witnesses to make their case against the president, making the trial much shorter than it might have been if they were allowed to present a longer case.

In recent weeks, Mr. Trump — who frequently calls Republican lawmakers to comment on their television appearances — has begun reaching out to them by inviting small groups of senators to the White House for lunch. While the meetings have not been billed as pertaining to a certain topic, the impeachment inquiry has come up. Mr. Trump has also had Republican lawmakers review transcripts of both of his phone calls with President Volodymyr Zelensky of Ukraine, before either call was released to the public for the first time.

Mr. Trump has also weighed in with his impressions of the House proceedings, and discussed strategy during the lunches.

“They’re just waiting for the House thing to play out,” said Senator Kevin Cramer, Republican of North Dakota, who attended a lunch last week.

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