Several people close to President Biden said on Thursday that they believe he has begun to accept the idea that he may not be able to win in November and may have to drop out of the race, bowing to the growing demands of many anxious members of his party.

One of the people close to him warned that the president had not yet made up his mind to leave the race after three weeks of insisting that almost nothing would drive him out. But another said that “reality is setting in” and that it would not be a surprise if Mr. Biden made an announcement soon endorsing Vice President Kamala Harris as his replacement.

This account is based on interviews with four people close to the president, all of whom described the situation as extremely delicate and spoke on the condition of anonymity to avoid offending the president. Mr. Biden remained in isolation at his vacation home in Rehoboth Beach, Del., after being diagnosed with Covid on Wednesday.

Many other Democrats more distant from the White House said expectations were rising within the party that the president would soon relent, a shift from just days ago when many were in despair about changing his mind. But there was also caution about reading signs from a president with an exceedingly small circle of confidants.

The latest prominent Democrat to publicly call on the president to consider passing the torch to another candidate was Representative Jamie Raskin of Maryland, a key member of the House committee that investigated the attack on the Capitol on Jan. 6, 2021. A letter he sent to Mr. Biden on July 6 and obtained by The New York Times on Thursday compared the 81-year-old commander in chief to a tiring baseball pitcher and urged him to consult with fellow Democrats about whether to continue his campaign. “Everything we believe in is on the line in the next four and a half months,” Mr. Raskin wrote.