Democratic leaders in the House and Senate refrained on Wednesday from urging their members to rally around President Biden and were instead listening to a barrage of complaints streaming in from both the party’s centrist wing and its progressives.

The private message from congressional leaders was that members should feel free to take whatever position about Mr. Biden’s candidacy was best for their districts, according to several people involved in the conversations who insisted on anonymity to discuss them.

For some, that meant joining the trickle of elected Democrats who have blasted the president’s performance and either called on him to withdraw or suggested he should seriously consider it.

Representative Raúl M. Grijalva, Democrat of Arizona, became the second Democrat in Congress to urge Mr. Biden to leave the race, following Representative Lloyd Doggett of Texas, who made his call on Tuesday.

“If he’s the candidate, I’m going to support him, but I think that this is an opportunity to look elsewhere,” Mr. Grijalva said in an interview. Referring to the president, Mr. Grijalva added: “What he needs to do is shoulder the responsibility for keeping that seat — and part of that responsibility is to get out of this race.”

Other Democrats were not ready to go that far, but hinted they were gravely concerned about Mr. Biden’s ability to serve and win re-election.

“I believe the president’s delivery at the debate was a disaster,” Representative Don Davis, a North Carolina Democrat in a competitive district, said in a statement on Tuesday. “Americans want a leader who is committed to telling the truth, but are also seeking a fighter.”

“President Biden needs to show that he is fit to lead the free world and demonstrate his fighting spirit,” Mr. Davis added. “If he’s going to stay in, he needs to step up.”

Several Democrats from politically competitive areas were awaiting the results of polls in their districts to decide whether they would join the calls for Mr. Biden to step aside. Members of Mr. Biden’s team — including Steve Ricchetti and Shuwanza Goff — were working the phones with lawmakers, trying to tamp down on the growing discontent.

But as congressional leaders called around to Democrats from battleground districts, they heard dire warnings, according to the people involved in the conversations. Some suggested there was no way Mr. Biden could defeat Mr. Trump. Others said Mr. Biden would drag House Democrats down. Still others urged Democrats to rally around Vice President Kamala Harris.

As swing district Democrats awaited the results of their own polls, The New York Times and Siena College published a survey showing that Mr. Trump’s lead in the race had widened after Mr. Biden’s fumbling debate performance. Mr. Trump now leads Mr. Biden 49 percent to 43 percent among likely voters nationally, a three-point swing toward the Republican from just a week earlier, before the debate.

At the same time, the Biden campaign circulated a memo to Democrats on Capitol Hill showing that its analysis of polling indicates little to no change in the race after the debate.

“Public polls and our own internal battleground top lines from last night show a steady race: We estimate that we’re down just one point. All of this is well within the margin of error and shows a steady race,” Hillary Beard of the Biden campaign wrote in an email to House Democrats.

Ms. Beard also touted a surge in fund-raising after the debate, saying the campaign raised more than $33 million. “The hour after the debate was the best grass roots fund-raising hour of the entire campaign,” she wrote.

Still, Democrats were uneasy. Their dilemma was illustrated by the actions of Senator Joe Manchin III of West Virginia, a longtime Democrat who recently abandoned the party to declare himself an independent. Mr. Manchin was so disillusioned by Mr. Biden’s debate performance last week — and angry that his phone calls to top Democrats to vent his alarm went unreturned — that he asked his staff to book him on several Sunday news talk shows to rail against the state of Mr. Biden’s campaign.

Eventually, Senator Chris Coons, Democrat of Delaware and a longtime ally of Mr. Biden, and other Democrats intervened with Mr. Manchin, who canceled the planned television appearances. The Washington Post reported earlier on Mr. Manchin’s change of plans.

Other lawmakers who have expressed openness to replacing Mr. Biden have also received calls from the campaign asking for more time to right the ship.

Senator Peter Welch, Democrat of Vermont, said he was leaving the door open to supporting a Democrat other than Mr. Biden.

“Let’s be candid: What we saw can’t be unseen,” Mr. Welch said on MSNBC of Mr. Biden’s faltering performance at last week’s debate.

“Age is the issue now,” Mr. Welch added. “That was the concern that folks had going into the debate. The debate was the opportunity to try to put that to rest, and candidly, it intensified the concerns about age.”

Representative Debbie Dingell, Democrat of Michigan, also told the news channel that Mr. Biden needed to make the case quickly to the American people that he has the stamina to do the job. She criticized Mr. Biden for not calling top Democrats in Congress the day after the debate.

“He does not have a lot of time,” Ms. Dingell said, adding: “He can’t be wrapped in bubble.”

Tim Balk contributed reporting.