President Biden and his advisers rushed to stem the first serious defections inside the Democratic Party since his shaky debate last week, as leading Democrats lent legitimacy to questions about his mental acuity and raised the specter of replacing him atop the ticket.

Mr. Biden’s operation hoped to assert fresh control on Wednesday, holding a call with a group of Democratic governors, in person and virtually, as he seeks to shore up support after days of private hand-wringing went public in sudden and quick succession.

On Tuesday, Mr. Biden suffered his first formal call to resign from the race from a Democratic member of Congress. The key Black lawmaker whose endorsement helped lift Mr. Biden to the nomination in 2020 said he would back the vice president if Mr. Biden “were to step aside.” And former House Speaker Nancy Pelosi said after Mr. Biden’s halting debate performance that it was “a legitimate question to say, ‘Is this an episode or is this a condition?’”

For days, the Biden campaign has insisted privately to donors and party activists and in memos that the race remains unchanged. But a private set of polls from a pro-Biden super PAC leaked to the news site Puck showed the president losing ground — around two percentage points — across all the most important battleground states. He was also now trailing in New Mexico, New Hampshire and Virginia, three states that were not seen a year ago as likely even to be contested seriously by Republicans.

The spate of early defections and diminished support in surveys demonstrates the scale of the crisis still gripping the Democratic Party. Though Mr. Biden’s aides have forcefully and repeatedly said publicly that the president has no plans to leave the race, the first public calls for him to step aside from elected lawmakers made clear that the matter remained far from settled.

The frustrations center on not only Mr. Biden’s dismal showing but also the actions he and his allies have taken since then to reassure Democrats that he is capable of winning the election. Many Democrats worry that Mr. Biden has moved too slowly to confront fears about his mental fitness and stamina, saying he should have done a series of interviews or campaign events in swing states almost immediately.

Efforts by the campaign and the White House to quell the concerns have created what amounts to a series of tests for the president in the coming days. They announced an interview with the ABC News anchor George Stephanopoulos on Friday, the president’s first in-depth appearance with a journalist in several weeks, as well as a news conference next week. At the end of the week, Mr. Biden will travel to Pennsylvania and Wisconsin for campaign events.

Any misstep in those appearances could ignite what has become a slow burn of conversations among groups of Democratic Party officials, top donors and elected leaders about whether the president should remain on the party’s ticket.

The discontent reflects how sharply Mr. Biden’s disastrous debate altered political expectations for Democrats. Fears about losing the White House have expanded well beyond the presidential race, fueling Democratic worries that Mr. Biden’s weakened standing — and the trajectory he has been on in polling for most of the year — could also eviscerate their chances of congressional control and of maintaining any toehold on power in Washington.

While top campaign and White House aides moved aggressively to assuage fears from supporters, Mr. Biden has been notably quiet in recent days. As of early Tuesday, he had yet to call either of the top two Democrats in Congress, Representative Hakeem Jeffries and Senator Chuck Schumer, according to two people familiar with the situation, but Mr. Biden called Mr. Jeffries at some point during the day. He has spent much of the past three days out of the public eye, emerging only to give brief remarks on Monday evening, and has not taken questions from reporters.

At a meeting on Monday of Democratic governors that was organized by Gov. Tim Walz of Minnesota, the chairman of the Democratic Governors Association, some of the party’s top elected leaders expressed frustration with the lack of contact with Mr. Biden since the debate, according to a person briefed on what took place. Most Democratic governors have not had direct contact with Mr. Biden since the debate, a fact that has caused exasperation and prompted continued questions about his health.

At the White House press briefing on Tuesday, Karine Jean-Pierre, the White House press secretary, faced a nearly hourlong barrage of questions about the president’s abilities, medications and physical examinations, including whether he had received a neurological test.

“We want to look forward,” said Ms. Jean-Pierre, who dismissed the idea of a neurological examination. “We will get out there across the country. Americans will see him for themselves.”

In a closed-door meeting this week with prominent Republican opponents of Mr. Trump, some participants, including former and current Republicans, told two top officials — Jeff Zients, the White House chief of staff, and Jen O’Malley Dillon, the campaign chair — that Mr. Biden should drop out of the race, according to two people familiar with what took place.

Mr. Biden’s aides responded by saying that was a nonstarter.

Still, comments by top campaign supporters have stoked speculation — and some measure of wishful thinking — that Mr. Biden could cede his spot on the ticket to Vice President Kamala Harris. On Tuesday, Representative James E. Clyburn of South Carolina, whose endorsement proved critical for Mr. Biden in the 2020 primary, told MSNBC that he still wanted a Biden-Harris ticket. But he also preemptively announced whom he would endorse if Mr. Biden departed the ticket.

“I will support her if he were to step aside,” Mr. Clyburn said of Ms. Harris.

Appearing outside a fund-raiser in San Francisco on Tuesday, Ms. Harris tried to tamp down talk of any ticket other than Biden-Harris, saying she was “proud to be Joe Biden’s running mate.”

“Look, Joe Biden is our nominee,” she told CBS News. “We beat Trump once, and we’re going to beat him again. Period.”

It is unclear whether efforts to rebuild confidence in the campaign will be successful, as jitters within the party seem to be growing by the day, spanning the breadth of Mr. Biden’s political coalition.

Two Democratic lawmakers who represent some of the most contested swing districts in the country — Representatives Jared Golden of Maine and Marie Gluesenkamp Perez of Washington — both offered public predictions that Mr. Trump would win the election. Senator Sheldon Whitehouse of Rhode Island openly asked for assurances from the White House about Mr. Biden’s “condition” in an interview with a local television station.

And Representative Lloyd Doggett of Texas, a veteran progressive lawmaker who represents an Austin-area district, released a statement calling on Mr. Biden to drop out of the race, becoming the first elected Democratic lawmaker to do so.

“President Biden saved our democracy by delivering us from Trump in 2020,” Mr. Doggett wrote. “He must not deliver us to Trump in 2024.”

Privately, many Democrats have said that they must now ensure they win the House to prevent Republicans from controlling all three branches of government, given the G.O.P.’s current advantage in Senate races and now the White House race. Some donors have begun discussing shifting funding from the presidential race to congressional contests, where they believe their party has greater chances of victory.

“You have to make sure you win the House, because the path to the House is the clearest one,” said Dan Sena, a former executive director of the House Democratic campaign arm.

One of the first polls released post-debate by an independent news organization, CNN, showed on Tuesday that Mr. Trump led Mr. Biden by six percentage points, 49 percent to 43 percent, which was consistent with his lead in previous CNN polls.

A majority of Democrats and Democratic-leaning voters said the party would have a better chance at winning if the nominee were someone other than Mr. Biden, according to the poll.

If Ms. Harris were the nominee, the race would be slightly closer, the poll showed, with her trailing Mr. Trump by two percentage points, 47 percent to 45 percent. Other Biden alternatives — Gov. Gavin Newsom of California, Transportation Secretary Pete Buttigieg and Gov. Gretchen Whitmer of Michigan — were also trailing.

Former Gov. David Walters of Oklahoma, a member of the Democratic National Committee, stopped short of calling on the president to step aside in an interview on Tuesday. But he said that if swing-state polling emerged showing other potential Democratic contenders in a much stronger position against Mr. Trump, then “that’s going to give everybody pause.”

“Donors, activists and others would respond a lot better if there was a more straightforward kind of updates and briefings about why it’s necessary to either continue on as we are, or otherwise,” he said. “You can overdo the rah-rah because people saw what they saw, and they know that 51 million people saw it as well.”

Appearing at a fund-raiser in McLean, Va., on Tuesday evening, Mr. Biden acknowledged anew his debate fumble but blamed it on a busy travel schedule in June that took him to Europe twice, though before the debate he spent nearly a week at Camp David resting and preparing.

“I came back and nearly fell asleep onstage,” Mr. Biden said.