Wealthy Democratic donors who believe a different nominee would be the party’s best chance to hold the White House are increasingly gritting their teeth in silence about President Biden, fearful that any move against him could backfire.

As of late Tuesday, the party’s moneyed class was carefully monitoring post-debate poll results and the positioning of elected Democrats for signs that support for Mr. Biden was cracking.

Earlier moves by donors to mount their own campaigns to pressure Mr. Biden to step down as the party’s presidential candidate have either fizzled out or prompted pushback from fellow contributors and operatives.

The deadlock reflects a broader paralysis within the party about how to handle a fraught situation that could inflame intraparty rifts, alienate key constituencies, damage personal relationships and benefit a Republican candidate most of the donors believe poses a threat to democracy.

The dynamic started taking shape mere hours after the debate.

At a breakfast on Friday morning at the Hotel Jerome in Aspen, Colo., where nearly 50 Democratic donors had gathered for a preplanned meeting convened by the super PAC American Bridge, one person asked the crowd for a show of hands of how many thought Mr. Biden should step aside. Nearly everyone in the room raised their hands, according to two people present.

Some members of the Democracy Alliance network of liberal financiers proposed a public statement calling on Mr. Biden to stand down, setting off a vigorous debate among some members of the group, with some floating their dream tickets. But the group’s board met after the debate and decided to maintain its support for Mr. Biden, according to a person briefed on the decision.

On a private email list including members of another liberal donor collective called Way to Win, participants expressed frustration with the Democratic Party’s circling of the wagons around Mr. Biden and urged that Vice President Kamala Harris be considered for the top spot on the ticket.

A small private online poll distributed after the debate to liberal donors and their advisers found that of dozens of respondents, more than 70 percent indicated that they were “ready to explore Plan B.”

And on Wall Street, some of Mr. Biden’s wealthiest past backers — including Seth Klarman, the billionaire chief executive of the hedge fund Baupost and a sharp critic of former President Donald J. Trump — have privately discussed whether to look beyond Mr. Biden, according to two people briefed on his thinking.

On Tuesday, both American Bridge and the Democracy Alliance hosted calls for donors anxious about the situation. Of the more than a dozen donors who spoke on the Bridge call, only one argued that the party should stick with Mr. Biden, according to a participant. Others expressed a desire to move on.

And next week, the dilemma is expected to be a hot topic on the sidelines of the annual meeting of chief executives at the Allen & Company summit in Sun Valley, Idaho, according to a person close to several major donors.

The deliberations among wealthy Democrats, detailed in more than two dozen interviews as well as in written communications reviewed by The New York Times, only intensified as the Biden campaign and the party establishment formed a protective wall around him in the days after the debate.

The party’s aggressive defense of Mr. Biden scared off donors from publicly calling for his replacement, said Maggie Kulyk, the owner of a wealth management firm and a board member of the Women Donors Network.

“Toeing this line makes us look almost, but not quite, as morally bankrupt as the Republican Party,” she said. “I mean, c’mon, man! Know when to say when.”

Ms. Kulyk added that donor coalitions might be timid about calling out Mr. Biden because they do not want to alienate donors “who feel strongly that we just need to stay the course.”

But, she said, “I don’t think the wall is very strong,” adding, “If a few voices came out, it could all come apart. And in my mind, I think that’s what needs to happen.”

Donor support is one of the indicators being closely watched to gauge whether Mr. Biden will be able to survive the mounting doubts caused by his weak debate performance. Other factors include the stances of elected Democrats, some of whom began on Tuesday to question whether it is in the party’s best interest to stick with Mr. Biden, as well as post-debate polls.

The backing of major contributors matters to Mr. Biden. Both Mr. Trump and Mr. Biden have harnessed digital fund-raising to their advantage during difficult moments, such as Mr. Biden’s catastrophic debate and Mr. Trump’s criminal conviction. But it is seven-figure donors who are essential to financing negative ads by super PACs, for instance.

The Biden team, along with some donors themselves, are uneasy about the suggestion that major contributors could orchestrate a change of the ticket.

“To suggest that the donor community could do that is scary,” said Craig Kaplan, a lawyer and Democratic donor in New York. “Money plays too much of a role in politics already.”

The debate has particularly riven Democratic donor groups that are typically fairly harmonious. At Way to Win, a donor collaborative founded during the peak of Trump-era resistance, organization leaders have touted Ms. Harris as a potential replacement for Mr. Biden.

Jen Fernandez Ancona, a founder of the group, stressed Ms. Harris’s electoral viability to allies in an internal email chain reviewed by The Times: “I absolutely believe we can do it with Harris at the top of a ticket and a good V.P. choice.”

Tory Gavito, another founder of Way to Win, said in an interview that the organization’s official position stopped short of calling for Ms. Harris to lead the ticket.

If Mr. Biden withdrew, and Ms. Harris ascended to the top of the ballot, she would inherit the campaign’s cash reserves, which stood at $212 million at the beginning of last month. If another candidate were elevated to the top of the ticket, things could get more complicated. “Think it through,” said Steve Silberstein, a major Democratic donor. “You’ve got to think three moves ahead of this game.”

Democratic donors have hotly debated prospective replacements beyond Ms. Harris, such as Gov. Gavin Newsom of California and Gov. Gretchen Whitmer of Michigan. Some donors have reached out to Mr. Newsom to encourage him to make a run, according to a person familiar with the outreach.

In Silicon Valley, efforts by Reid Hoffman, the billionaire founder of LinkedIn, to rally fellow Silicon Valley donors behind Mr. Biden have been controversial among his peers. In an email blast this week, a Hoffman adviser criticized Way to Win and Democracy Alliance.

One billionaire megadonor peer of Mr. Hoffman called an email he had sent minimizing the impact of the debate “shocking.” And some other donors who received it found it too dismissive of the risks of keeping Mr. Biden as the nominee, according to two people close to other major donors. Tech entrepreneurs who have been dismissive of Mr. Biden’s chances privately include the investor Ron Conway and the StubHub chief executive Eric Baker, according to people close to them.

On Wall Street, where executives tend to be unsentimental about cutting their losses, a half-dozen prominent Democratic donors said Mr. Biden’s chances had plummeted after the debate, at times using expletives to describe the situation.

In recent days, some company leaders and Democratic political operatives have called the chief executive of JPMorgan Chase, Jamie Dimon, according to someone with knowledge of the calls, to gauge his interest in replacing Mr. Biden. (Mr. Dimon has said repeatedly that he will not run.)

Michael Novogratz, a cryptocurrency investor who supported Mr. Biden in 2020 but now worries about his age, welcomed the moment. “I have been advocating for other choices,” he wrote in a text to The Times. “The poor performance at the debate opened that door for the first time.”

Mr. Klarman, the hedge-fund manager and a strong supporter of Mr. Biden in the run-up to the debate, has in recent days told friends and fellow donors that they should seriously consider exploring a Plan B, citing his debate performance, according to a person who has spoken to him.

“The most important thing President Biden — and all of us who have supported him to date — can do is prioritize the defeat of Donald Trump in this election,” Mr. Klarman said in a statement. “I trust President Biden, who has been a truly great president, will continue to keep this at the center of every decision about the path forward.”

Some donors and their political advisers said in interviews that Democrats would be better off focusing on groups that could help the party regardless of who is atop the ticket.

Steve Phillips, a longtime Democratic donor in San Francisco who described himself as having been “deluged with frantic calls, emails and texts from fellow Democratic donors desperately seeking a way to remove Biden from the presidential ticket,” said he had counseled his peers to “just hunker down, ride this out and focus on voter mobilization.” He added, “That’s a much better use of your time than worrying about whether Gretchen Whitmer could be the candidate.”

Donors pushing for replacing Mr. Biden should be careful what they wish for, warned John Morgan, a lawyer who said he had raised nearly $1 million for the Biden campaign and was planning a summer fund-raiser for Mr. Biden.

“I fear that the scrum for a new nominee could cause more infighting and do more harm than good in the overall scheme,” Mr. Morgan wrote in a text. “None of us will decide — the president will.”

Still, the president’s supportive outside groups and their donors are beginning to plot how they would shift gears without Mr. Biden. The billionaire Facebook co-founder Dustin Moskovitz, one of the party’s single biggest givers, was asked on social media on Tuesday what the party should do about Mr. Biden.

“Ah well that’s easy, they just need to,” Mr. Moskovitz joked, cutting himself off midsentence. He did not write another word.