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President Biden found himself increasingly isolated on Thursday as a small group of his longtime aides and advisers have become convinced that he will have to make what they see as the painful but inevitable decision to abandon his campaign for re-election, according to three people who have been briefed on the matter.

In recent days, the group has been trying to come up with ways to persuade Mr. Biden to step aside from the campaign. Those discussions were recounted by three people familiar with them who, like others in this article, spoke on the condition of anonymity to discuss a sensitive situation. There is no indication that any of the discussions have reached Mr. Biden himself, one of the informed people said.

The effort comes after a disastrous debate performance in Atlanta two weeks ago plunged Mr. Biden’s candidacy into crisis. He is facing pressure from all sides: Lawmakers on Capitol Hill are vacillating between tepid support and outright calls for him to drop out, and those calls were being echoed by some of the wealthiest donors in the party.

At the center of it all is Mr. Biden, who is convinced that he is the only one who can beat former President Donald J. Trump, according to people familiar with this thinking, and who is unlikely to be persuaded otherwise by people outside his insular inner circle, which includes his family.

He faced a new test on Thursday night in a news conference following the NATO summit in Washington. In an early stumble before it even got underway, Mr. Biden flubbed his introduction of President Volodymyr Zelensky of Ukraine, saying: “Ladies and gentlemen, President Putin.” He quickly caught himself.

During the news conference, he referred to “Vice President Trump” when he meant Vice President Kamala Harris, a mistake that former President Donald J. Trump immediately mocked on social media.

But Mr. Biden showed a command of the issues on foreign policy, although he spoke slowly and meandered at times. Lawmakers and aides in Congress said it was a strong enough performance to keep the dam from breaking with mass calls for Mr. Biden to step aside, but with enough missteps to prolong the anxiety on Capitol Hill.