“I’m not a racist. I am the least racist person you have ever interviewed, that I can tell you,” Mr. Trump said as he arrived at Trump International Golf Club for dinner with Representative Kevin McCarthy of California, the majority leader.

The rift over Mr. Trump’s comments, and how they have since been recounted, risked further eroding trust between Democrats and Republicans at the beginning of a critical week for Congress. Government funding is set to expire on Friday, and lawmakers will need to pass a stopgap spending measure to avoid a government shutdown on Saturday.

And lawmakers are already facing a difficult fight over the politically volatile subject of immigration, with the fates of hundreds of thousands of young immigrants hanging in the balance. Adding to the uncertain picture for those immigrants, the Trump administration resumed accepting renewals for the program over the weekend, under orders from a federal judge who is hearing a legal challenge to Mr. Trump’s dismantling of the program.

But in Congress, the battle took on an increasingly personal dimension as Mr. Perdue and Mr. Cotton essentially accused Mr. Durbin of lying about the president’s comments, even after the vulgar remarks were widely reported and the White House did not immediately dispute that the president had made them.

“I didn’t hear that word either,” Mr. Cotton said on CBS’s “Face the Nation.” “And I was sitting no further away from Donald Trump than Dick Durbin was.”

Mr. Cotton said Mr. Durbin “has a history of misrepresenting what happens in White House meetings,” an assertion that Mr. Perdue made in his own interview Sunday morning on ABC’s “This Week.”

Ben Marter, a spokesman for Mr. Durbin, responded by suggesting that Mr. Perdue and Mr. Cotton should not be believed.

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“Credibility is something that’s built by being consistently honest over time,” Mr. Marter wrote on Twitter. “Senator Durbin has it. Senator Perdue does not. Ask anyone who’s dealt with both.”

Mr. Graham had previously told a fellow South Carolina Republican, Senator Tim Scott, that reports in the news media of Mr. Trump’s language were “basically accurate.” A spokesman for Mr. Graham did not respond to a request for comment on Sunday.

Senator Jeff Flake, an Arizona Republican who is part of a bipartisan group of senators that has developed an immigration proposal, said on Sunday that people in the room with Mr. Trump during Thursday’s meeting told him that the president had used the inflammatory language.

“I was in a meeting directly afterwards where those who had presented to the president our proposal spoke about the meeting,” he said on “This Week.” “I heard that account before the account even went public.”

The other lawmakers at the meeting, all Republicans, have not offered any public recollection of what the president said.

The Homeland Security secretary, Kirstjen Nielsen, who also attended the meeting, said on “Fox News Sunday” that she did not recall the president “saying that exact phrase.”

Mr. Durbin had told reporters on Friday that Mr. Trump called African nations “shitholes,” which Mr. Durbin said was “the exact word used by the president, not just once, but repeatedly.” He called the president’s comments “hate-filled, vile and racist.” At the meeting, Mr. Durbin said Mr. Trump also questioned whether the United States needed more Haitians.

Mr. Graham is said to have admonished the president during the meeting, telling him that “America is an idea, not a race.”

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Saying that President Barack Obama had exceeded his authority when he created the program that shields from deportation young immigrants who were brought to the United States illegally as children, known as Dreamers, Mr. Trump moved to end it in September.

He gave Congress six months to find a fix for the program, called Deferred Action for Childhood Arrivals, or DACA.

Democrats have been pushing to secure a deal by Friday’s government funding deadline that would protect Dreamers, hoping to capitalize on the leverage they have as a result of that deadline. Democratic votes will be needed to pass the stopgap spending measure in the Senate, where government funding measures require 60 votes, and Democratic votes might be needed in the House as well.

Republican leaders say they want to address DACA as well, but separately from funding the government. Compared with their Democratic counterparts, Republican leaders are operating on a longer time frame for taking action, given the six-month window that Mr. Trump gave Congress. They also have to contend with internal divisions over immigration policy.

The bipartisan group of senators, including Mr. Durbin and Mr. Graham, reached an agreement last week that would provide a path to citizenship for DACA recipients while also providing money for border security and making other changes to immigration policy.

But Mr. Trump dismissed the proposal, calling it a “big step backwards.” And on Sunday, he offered a pessimistic take on Twitter: “DACA is probably dead because the Democrats don’t really want it, they just want to talk and take desperately needed money away from our military.”

Still, administration officials said they intended to abide by an order from Judge William Alsup of Federal District Court in San Francisco last week to restart the DACA program, with some modifications, while a legal challenge plays out. On Saturday, officials did just that by updating the program’s website to once again accept renewals.

But administration officials hope the judge’s decision will be temporary. Officials said the president’s lawyers are examining whether to appeal his order, which could lead to a ruling allowing the administration to shut the program down again. The administration could also choose to modify its legal reasoning to satisfy the judge’s criticisms.

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Either way, immigrant rights activists are not counting on legal action to be the ultimate protection for the Dreamers. Several said they believe the only real solution for the hundreds of thousands of young immigrants is to convince Congress to act soon.

The court ruling could lessen the pressure for that kind of action — at least in the short term — since some young immigrants can once again renew their protected status for another two years.

Lawyers and directors of community legal services spent Sunday preparing fact sheets and answering calls that have been flooding their offices.

Most of the calls that Hasan Shafiqullah, the director of the immigration unit of the Legal Aid Society of New York, said he has been receiving started with the burning questions “Is this real? Can I file?”

The answer, for now, he said, is yes. But he is concerned for his clients about another turnabout in the courts.

“It’s just the emotional roller coaster that our clients are on,” he said.

Allan Wernick, the director of CUNY Citizenship Now, a legal services program at the City University of New York, said filing renewals could be very powerful, at least symbolically, to Mr. Trump.

“The more applications get in, the more it is clear that his ending the program has real-world impact,” he said.

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