In a sign of the conflicting pressures surrounding Mr. Biden, the vice president has told people that the terminal brain cancer of Beau Biden, who died in May, had caused him to consider resigning the vice presidency to take care of his grieving family, though those aware of the vice president’s thinking say that idea never became too serious.

David Axelrod, a former Obama adviser who has spoken with Mr. Biden, said what many were thinking about the feasibility of his candidacy.

“I understand completely why he would consider running,” said Mr. Axelrod, pointing to Mr. Biden’s 36 years in the Senate and two terms at Mr. Obama’s side. “On the other side is the reality of running for president. The fund-raising, the demands of campaigning and organizing, the constant and irritating exposure and the prospect of running against a well-fortified opponent who has a huge head start. Add all that up, and it is a counterweight to why he should run.”

Senator Richard J. Durbin, Democrat of Illinois, said Mr. Biden appreciated “better than any candidate in the race” the rough-and-tumble nature of presidential campaigns. “He’s been through this before; he understands what is involved.”

But Mr. Durbin contended that legacy should not be part of Mr. Biden’s consideration because “the reputation he enjoys in this country is solid, whatever his final decision.”

Some of his donors agreed and have been encouraging, to a point.

“He just has to look inside himself and decide, and people like me will support whatever decision he makes,” said Richard Davis, a donor to and supporter of Mr. Biden in New York. “But I only want him to proceed if he concludes he has a reasonably good chance of actually winning.”

But others say that Mr. Biden, who struggled in White House races in 1988 and 2008, should not be reluctant to jump in, and they noted that he showed a command of the issues and performed strongly in debates that have become increasingly important presidential campaign moments.