WASHINGTON — President Trump appeared to secure enough support on Monday to fill the Supreme Court seat left open by the death of Justice Ruth Bader Ginsburg, although Senate Republicans remained publicly undecided if they would try to force through his nominee before the election or wait until after voters have decided whether to grant him a second term.

With key Republican senators agreeing to install a new justice at least by the end of this year, Trump said he would announce his choice for the seat by Friday or “probably Saturday,’’ after memorial services for Ginsburg, and pressed his allies to vote before the election in what would be the fastest contested Supreme Court confirmation in modern history.

Such a timetable, however, would leave only 38 days for the Senate to act and, as a practical matter, even less time because it is highly unlikely that Republicans would want to vote in the last few days before an election in which several of them face serious threats. Some senior Republican senators were still expressing caution about such an accelerated timetable even with the votes seemingly in hand.

The president was buoyed after Senators Charles Grassley of Iowa and Cory Gardner of Colorado, two of three remaining Republicans who might have opposed filling the seat, announced that they would support moving ahead with a nomination even though they refused to consider President Barack Obama’s nomination in an election year in 2016. That left only Senator Mitt Romney of Utah considered undecided, but even without him, it appeared to guarantee at least 50 Republican votes to move ahead, with Vice President Mike Pence available to break a tie.

With polls showing Trump trailing former Vice President Joe Biden, the Democratic presidential nominee, the president insisted on pressing ahead without waiting for the election. “I’d much rather have a vote before the election because there’s a lot of work to be done, and I’d much rather have it,’’ Trump told reporters. “We have plenty of time to do it. I mean, there’s really a lot of time.’’

Trump privately met at the White House on Monday with Judge Amy Coney Barrett of the 7th US Circuit Court of Appeals, in Chicago, his front-runner and a favorite of anti-abortion conservatives.

The politics of Ginsburg’s replacement roiled Washington as senators returned to town for the first time since her death. Two Republican senators, Susan Collins of Maine and Lisa Murkowski of Alaska, said over the weekend that they opposed filling the seat until voters decide the presidency.

But Senator Mitch McConnell of Kentucky, the Republican leader, reiterated that he intended to fill the seat before year’s end, without explicitly committing to a vote before the election.

“This Senate will vote on this nomination this year,’’ he said in a speech that was intended to justify proceeding after Republicans refused to even consider Obama’s nomination of Judge Merrick Garland for almost nine months in 2016 partly on the grounds that voters should have a say in who filled the lifetime appointment.

McConnell and other Republicans rationalized taking the opposite position this year because their party controls both the White House and the Senate.

Senator Lindsey Graham of South Carolina, the chairman of the Judiciary Committee that will consider the president’s nomination, had vowed repeatedly not to support confirming any selection by Trump in an election year in keeping with the 2016 decision, only to flip-flop during the weekend.

In a letter to Democrats Monday, Graham said he reversed himself in retaliation for the Democrats’ treatment of Justice Brett Kavanaugh when he was confirmed in 2018 and because Republicans have the power to proceed. “I am certain if the shoe were on the other foot, you would do the same,’’ Graham wrote.

Romney, a frequent critic of Trump, said he planned to announce his views after a senators’ lunch on Tuesday.

Senator Chuck Schumer of New York, the Democratic leader, excoriated Republicans for what he called a brazen power play. “To try and decide this at this late moment is despicable and wrong and against democracy,’’ Schumer told reporters.

As Barrett met with the president on Monday, people close to the process said she remained the likeliest choice, but there was still attention on Judge Barbara Lagoa of the 11th Circuit because she is a Cuban-American from Florida, a critical state for the president’s reelection chances.

Trump told reporters that he had narrowed the list to five women, but the other three identified by the people informed about the process were seen as long shots: Kate Todd, a deputy White House counsel, and Judges Allison Jones Rushing of the 4th Circuit and Joan Larsen of the 6th Circuit.

But even as he talked about showing respect for Ginsburg, Trump asserted with absolutely zero evidence that her dying wish that she not be replaced until the next president is chosen, as conveyed by her granddaughter to NPR, was actually scripted by Democrats like House Speaker Nancy Pelosi, Schumer or Representative Adam Schiff of California.

“I don’t know that she said that, or was that written out by Adam Schiff and Schumer and Pelosi?’’ he told “Fox & Friends.’’