Former Gov. Andrew M. Cuomo told donors at a private fund-raiser in the Hamptons last weekend that he believed President Trump would wade into the race for New York City mayor and help clear a path for his election, according to audio obtained by Politico.

Mr. Cuomo, who is running as an independent in the general election after losing June’s Democratic primary, made the comments on Saturday as he sought to convince Trump-friendly donors that he was best positioned in the crowded field of candidates to defeat Assemblyman Zohran Mamdani, his party’s leftist nominee.

Mr. Cuomo predicted, without citing evidence, that Mayor Eric Adams, another Democrat running as an independent, would bow out of the contest “at the end of the day,” rather than be “a spoiler.” Mr. Cuomo quipped that if Mr. Mamdani won, and Mr. Adams was blamed, the mayor would “have to move to Florida.”

The former governor said he expected that Curtis Sliwa, the Republican nominee, would stay in the race, but that Mr. Trump and other Republican leaders would urge their supporters not to vote for him.

“I think we can minimize his vote, because he’ll never be a serious candidate,” Mr. Cuomo said of Mr. Sliwa, according to Politico. “And Trump himself, as well as top Republicans, will say the goal is to stop Mamdani. And you’ll be wasting your vote on Sliwa.”

“So I feel good about that,” he added.

The Times did not independently obtain audio of the fund-raiser, but Politico published a transcript of the remarks.

Two prominent political donors, who did not attend Saturday’s event and insisted on anonymity, separately told The Times that Mr. Cuomo and his aides had made similar comments to them in recent conversations.

Rich Azzopardi, a spokesman for Mr. Cuomo, did not dispute Mr. Cuomo’s remarks as reported by Politico, but said in a statement that Mr. Cuomo had been speaking hypothetically. He pointed out that Mr. Trump had not endorsed Mr. Sliwa, even though he was his party’s nominee.

“We’re not asking for or expecting help from anyone,” Mr. Azzopardi said.

The revelation of the private comments on Tuesday made waves in New York and Washington at a time when Mr. Trump is said to be weighing whether to intervene in the race or potentially strike an alliance with Mr. Cuomo, a fellow son of Queens whom he has known for decades.

If he were to do so publicly, it could cause headaches for Mr. Cuomo: New York is an overwhelmingly Democratic city where Mr. Trump is unpopular and Mr. Mamdani remains the favorite to win in November. But polls show the former governor has almost no chance of becoming mayor as long as the votes of people who oppose Mr. Mamdani are split between him, Mr. Adams and Mr. Sliwa.

The New York Times reported earlier this month that Mr. Trump and Mr. Cuomo spoke directly about the race as the president mulled whether to get involved. Though the call was described to The Times by three people briefed on it, both men denied that it had taken place.

But when Mr. Cuomo was asked by an attendee at the event on Saturday whether he and his team were talking to the White House, he did not say no.

“Let’s put it this way: I knew the president very well,” Mr. Cuomo said, according to the report. He added, “I also believe that he’s not going to want to fight with me in New York if he can avoid it.”

The fund-raiser in the moneyed summer enclave of Southampton was hosted by Jimmy Finkelstein, a conservative news executive, and his brother, Andrew Stein, a former New York City Council president and a decades-long friend of Mr. Trump.

The Times reported that Mr. Stein and the pollster Mark Penn recently met with Mr. Trump at the White House and shared polling that showed that Mr. Cuomo was the strongest competitor to Mr. Mamdani.

It was not clear whether Mr. Cuomo’s comments on Saturday were informed by conversations with the White House or if they were an attempt to sway donors to help him pressure his rivals out of the race.

The White House did not immediately respond to a request for comment.

Mr. Mamdani, for his part, sought to capitalize on his rival’s apparent openness to Mr. Trump’s intervention.

“Since he’s too afraid to say it to New Yorkers’ faces, we’ll make it clear,” said Dora Pekec, a spokeswoman for Mr. Mamdani. “Andrew Cuomo is Donald Trump’s choice for mayor.”

Questions about whether any of the candidates will drop out voluntarily — or could be induced to do so with offers of other jobs — have swirled for months among political operatives and some wealthy donors who are alarmed by Mr. Mamdani’s democratic socialist views.

Both Mr. Adams and Mr. Sliwa have repeatedly insisted they are not going anywhere. Mr. Cuomo has likewise rejected calls to step aside.

In a statement on Tuesday, Mr. Adams said Mr. Cuomo “is the only one to blame for this mess” and called on him again to drop out.

“The truth is he’s an embarrassing double-digit loser who couldn’t beat a socialist, and his failure created this entire scenario,” the mayor said.

Mr. Sliwa, speaking on WABC radio, mocked Mr. Cuomo for holding a spree of fund-raisers in the Hamptons over the weekend and no campaign events in New York City. Mr. Sliwa said it was a fantasy to believe his supporters would ever vote for Mr. Cuomo, who enacted liberal policies in Albany, including ending cash bail for some offenses.

“Ain’t no Republicans voting for you, no-bail Cuomo,” Mr. Sliwa said.

In public appearances, Mr. Cuomo has sought to portray himself as the only candidate capable of going head-to-head with Mr. Trump, whose administration he has said is trying to “undermine our democracy.”

But he has used a different tone behind closed doors. Earlier this month, he told business leaders that “personally” neither he nor Mr. Trump wanted a fight with the other.

On Saturday, Mr. Cuomo also at least entertained means of clearing the field that did not involve Mr. Trump.

After an attendee asked “if it’s legal or ethical to offer Eric Adams something” to leave the race, Mr. Cuomo quipped that Mr. Stein, who pleaded guilty to tax evasion in 2010, was working on it, according to Politico.

“It is neither legal nor ethical,” Mr. Cuomo said to laughter. “But Andy is talking to him.”

Mr. Stein declined to comment.

Benjamin Oreskes and Maggie Haberman contributed reporting.