With President Biden under pressure to drop his bid for a second term, his party has been thrust into uncharted territory, struggling with a long list of risks and rewards as it faces the prospect of replacing Mr. Biden less than two months before the party convention.

No presumptive nominee has withdrawn this late in the process. But no party has faced the challenge the Democrats face today: a nominee dogged by doubts about his mental acuity; his ability to beat his rival, former President Donald J. Trump; and his fitness to serve another four years as president.

All of this has left Democrats struggling with critical questions: Is it easier to defeat Mr. Trump with or without Mr. Biden at the top of the ticket? Is it riskier to go with a new candidate or stick with a president who appears headed for defeat?

On Wednesday, a New York Times/Siena College poll found that Mr. Trump’s lead over Mr. Biden among likely voters had grown to six percentage points after the president’s halting debate performance last week.

The White House said the president was not dropping out, and he met with Democratic governors on Wednesday. But he confided to at least two allies that he realized the next few days were crucial to saving his candidacy. To that end, Mr. Biden began preparing for his first sit-down interview since the debate, with George Stephanopoulos of ABC News, on Friday morning.

Several Democrats said that no matter the risks, a new nominee could bring a host of benefits to the party, particularly if Mr. Biden anointed a successor in an effort to assure a smooth transition and minimize intraparty battling.