One Democrat with direct knowledge of the conversations described the outreach as a heady combination of donors and friends of Mr. Biden’s wanting to prop him up in his darkest hours, and of recent polls showing Mrs. Clinton’s support among independents declining, suggesting there could be a path to the nomination for the vice president.

Ms. Dowd reported that as Beau Biden faced brain cancer, he “tried to make his father promise to run, arguing that the White House should not revert to the Clintons and that the country would be better off with Biden values.” Mr. Biden’s other son, Hunter, also encouraged him to run, she wrote.

The support Mr. Biden has garnered speaks to growing concerns among Democrats that Mrs. Clinton could lose in Iowa and New Hampshire, as the populist message of one of her opponents, Senator Bernie Sanders of Vermont, draws swelling crowds.

“The reality is it’s going to be a tough, even-steven kind of race, and there’s that moment when a lot of party establishment would start exactly this kind of rumble: ‘Is there anybody else?’ ” said Joe Trippi, a Democratic strategist.

At the same time, the slow trickle of news about Mrs. Clinton’s use of private email when she was secretary of state, and the coming Benghazi hearings, may be distracting some voters from the core message of her campaign: the need to lift the middle class.

“It’s not that we dislike Hillary, it’s that we want to win the White House,” said Richard A. Harpootlian, a lawyer and Democratic donor in Columbia, S.C., who met with Mr. Ricchetti before Beau Biden died. “We have a better chance of doing that with somebody who is not going to have all the distractions of a Clinton campaign.”

A spokeswoman for the Clinton campaign declined to comment.

In a July 30 Quinnipiac poll, 57 percent of voters said Mrs. Clinton was not honest and trustworthy, and 52 percent said she did not care about their needs or problems. The same poll showed Mr. Biden with his highest favorability rating, 49 percent, in seven years, with 58 percent saying he was honest and trustworthy and 57 percent saying he cared about them. But Mrs. Clinton’s numbers are still strong, especially among likely Democratic primary voters.