President Biden’s family is urging him to stay in the race and keep fighting despite last week’s disastrous debate performance, even as some members of his clan privately expressed exasperation at how he was prepared for the event by his staff, people close to the situation said on Sunday.

Mr. Biden huddled with his wife, children and grandchildren at Camp David while he tried to figure out how to tamp down Democratic anxiety. While his relatives were acutely aware of how poorly he did against former President Donald J. Trump, they argued that he could still show the country that he remains capable of serving for another four years.

Mr. Biden has been soliciting ideas from advisers about how to proceed, and his staff has been discussing whether he should hold a news conference or sit for interviews to defend himself and change the narrative, but nothing has been decided yet. The campaign scheduled what could be a critical call with its national finance committee for Monday to calm nerves and take temperatures.

One of the strongest voices imploring Mr. Biden to resist pressure to drop out was his son Hunter Biden, whom the president has long leaned on for advice, said one of the people informed about the discussions, who, like others, spoke on condition of anonymity to share internal deliberations. Hunter Biden wants Americans to see the version of his father that he knows — scrappy and in command of the facts — rather than the stumbling, aging president Americans saw on Thursday night.

Other family members were trying to figure out how they could be helpful. At least one of the president’s grandchildren has expressed interest in getting more involved with the campaign, perhaps by talking with influencers on social media, according to the informed person.

One of the people informed about the situation said “the entire family is united” and added flatly that the president was not getting out of the race and had not discussed doing so. “You get up and keep fighting,” the person said.

The anger among Democrats was made evident on Sunday when John Morgan, a top Democratic donor who is close to Mr. Biden’s brother Frank, publicly blamed the advisers who managed the president’s debate preparations, citing by name Ron Klain, Anita Dunn and Bob Bauer.

“Biden has for too long been fooled by the value of Anita Dunn and her husband,” Mr. Morgan wrote on social media. “They need to go … TODAY. The grifting is gross. It was political malpractice.”

He elaborated in a subsequent interview. “It would be like if you took a prizefighter who was going to have a title fight and put him in a sauna for 15 hours then said, ‘Go fight,’” he said. “I believe that the debate is solely on Ron Klain, Bob Bauer and Anita Dunn.”

They were asking why Mr. Klain, the former White House chief of staff who ran the preparations, would in their view allow him to be overloaded with statistics, and they were angry that Mr. Biden, who arrived for the debate in Atlanta with a summer tan, was made up to look pale and pallid, said one of the people, who has been in touch with several members of the family.

But the person said that the president himself was not among those who were upset and that he still trusted Mr. Klain, Ms. Dunn, Mr. Bauer and the others. A person close to Jill Biden, the first lady, said she was not critical of them, either. A White House official and a second person close to the family later denied that other members were mad. Several Democrats said it was unfair to blame the staff for the president’s own failings, dismissing what they called typical second-guessing and scapegoating aimed at diverting fault away from Mr. Biden himself.

A couple of Democrats pointed out that neither family members nor Mr. Morgan or other critics attended the preparation sessions and therefore had no idea how they went. One member of Mr. Biden’s circle said that no one was happy with how the debate turned out and that it was human nature to look for someone to blame.

Mr. Klain, Ms. Dunn and Mr. Bauer had no comment about the debate preparation, but Mr. Klain said that it was 100 percent certain the president would stay in the race.

“He is the choice of the Democratic voters,” Mr. Klain said. “We are seeing record levels of support from grass-roots donors. We had a bad debate night. But you win campaigns by fighting — not quitting — in the face of adversity.”

He recalled a primary debate in 2019 that went badly but did not stop Mr. Biden. “It’s a tough, close campaign and he’s the person who can win it,” Mr. Klain said. “Big-money donors don’t get to dictate the nominee of the Democratic Party.”

In the days since the debate, Mr. Biden has privately and publicly acknowledged that he did not do well, and he has been calling trusted advisers like Mr. Klain; Ted Kaufman, his longtime aide and friend; and Jon Meacham, the historian and informal adviser; as well as key donors and party figures.

But three people familiar with Mr. Biden’s calls said that they were more about checking in to see what people were saying, rather than to seek advice about reassessing his future. His tone was described as measured. One of the people on Mr. Biden’s phone tree said that the president wanted to keep campaigning hard to drive a contrast with Mr. Trump, a convicted felon who tried to overturn the last election and made numerous false statements during the debate.

Campaign advisers have been burning up the phone lines all weekend with major donors angry about the situation in hopes of heading off a wave of defections. The campaign scheduled a conference call for 5:30 p.m. on Monday for its national finance committee to hear from Jen O’Malley Dillon, the campaign chair. Many insiders have said that preserving the donor base will be key to the president staying in the race.

Mr. Biden is scheduled to return to the White House on Monday evening and has a relatively light public schedule for the week. On Tuesday, he plans to visit an emergency operations center to discuss extreme weather and will headline an evening fund-raiser in McLean, Va. He will host a Medal of Honor ceremony on Wednesday and a Fourth of July barbecue for military service members on Thursday before heading to Delaware on Friday. A cabinet meeting set for Wednesday was canceled, according to officials, who said too many members were traveling the day before the holiday.

While the campaign has forcefully rejected advice that Mr. Biden step aside for another candidate just weeks before the roll call vote to formalize his nomination, many Democrats, including some working for the president, said they did not think the door was yet closed on that possibility.

But Mr. Biden is a proud man, and they said they believed that the odds of him trying to gut it out were still 4 or 5 to 1. The only way they said they could imagine him reversing course was if he could be afforded a dignified way out in which he could claim credit for ousting Mr. Trump in 2020, restoring the country and serving as a transition to the next generation.

A new poll by CBS News found strong sentiment among Democratic voters for Mr. Biden, 81, to cede the way to a younger nominee. Forty-five percent of Democrats said they wanted a different candidate to take on the battle with Mr. Trump. Among voters overall, just 27 percent think Mr. Biden has the mental and cognitive health to serve as president, down from 35 percent before the debate.

Democratic allies took to the Sunday talk shows to defend the president. “If they weren’t engaged in a little bit of hand-wringing, they wouldn’t be Democrats,” Senator Raphael Warnock of Georgia said on “Meet the Press” on NBC. But he added, “Joe Biden has demonstrated, not over 90 minutes, but over the last four years, the character and the mettle of the man that he is.”

Gov. Wes Moore of Maryland acknowledged that Mr. Biden’s age was a concern for voters. “The number 81 is an important number,” he said on “Face the Nation” on CBS. “But so is watching historically low unemployment rates. And I don’t think that people should lose sight of that.”

Mr. Moore said he would not run if Mr. Biden did drop out. “Joe Biden is not going to take himself out of this race, nor should he,” he said. “He has been a remarkable partner.”

Former Speaker Nancy Pelosi dismissed talk of the president dropping out. “I support the Biden-Harris ticket,” she told Jen Psaki, a former Biden White House press secretary, on MSNBC. “I’m not abandoning Joe Biden right now, for any speculation.”

The “right now” in that comment, however, did not go unnoticed, and Democrats were still watching to see what their senior elected leaders would do, wondering whether they might intervene privately with the president despite their public comments of support. Ms. Pelosi’s spokesman later said that she “has full confidence” in Mr. Biden and that “any suggestion otherwise is not based in fact.”

Representative Jamie Raskin, Democrat of Maryland, though, acknowledged that the president’s fate was uncertain. “There are very honest and serious and rigorous conversations taking place at every level in our party,” he said on MSNBC, adding that the party would be unified “whether he’s the candidate or someone else is the candidate.”

If any major discussions about the president’s future were to take place with the family, two Biden confidants said, they would not happen at Camp David, where too many people outside the family might overhear.

The family had already planned before the debate to spend the weekend at Camp David, in part to participate in a photo shoot with the veteran celebrity photographer Annie Leibovitz. It was the first time the entire family had assembled in one place since Hunter Biden was convicted of federal gun charges; he still faces sentencing and another trial on tax charges.

A senior administration official who was not authorized to detail internal conversations said there was an ongoing debate over how the president moves forward — not about dropping out, but about how best to make the case that he should not.

The version of Mr. Biden that has emerged in rallies and at fund-raisers since Atlanta is more in line with the person his aides describe — someone who is energized, emphatic and willing to keep fighting until November.

But some aides were not happy to see him relying on a teleprompter in fund-raisers, a practice pushed by advisers seeking a more disciplined approach by the president even in informal settings. One aide said that Mr. Biden had been “scared” away from a more informal approach in recent months.