But the vice president and his closest advisers are also monitoring Mrs. Clinton as she tries to put questions over her use of a private email server while secretary of state behind her. On Thursday, Quinnipiac University released a poll showing troubling signs for Mrs. Clinton in three key states: Florida, Pennsylvania and Ohio. In each state survey, at least 6 in 10 voters said the former secretary of state was not honest and trustworthy. Fewer than 4 in 10 voters in each state held a favorable view of Mrs. Clinton, compared with nearly 6 in 10 who viewed her negatively.

Those who have spoken directly to Mr. Biden, who spent part of this week with his grandchildren in Delaware, increasingly believe that he is going to take several more weeks to decide whether to pursue a candidacy. They hope the time will provide more clarity about how much political damage Mrs. Clinton has suffered as a result of the email issue, and whether the problems will last.

Mr. Biden is still mourning the death of his son Beau, and his aides had previously said that he would make an announcement about whether he will run by the end of this summer. But it now appears that deadline could slip. He would need to make a declaration before the first Democratic debate on Oct. 13.

“He’s not unrealistic that if she’s able to right the ship here, at this point it’s tough, really tough, to see a lot of Democrats coming off her,” said one former senior Obama administration official who recently discussed the merits of a campaign with Mr. Biden. “On the other hand, the things with the emails and everything else could deteriorate — who knows? So I think he’s figuring he’s got another month or so to see what happens, to sort of put feelers out there.”

Kendra Barkoff, a spokeswoman for Mr. Biden, declined to comment on questions about calls that have been made and the efforts of the vice president’s allies.

Members of Mr. Biden’s inner circle have asked the Democrats they have spoken with to be circumspect about conversations, but the advisers have followed who is saying favorable things in the news media about the vice president’s prospects, and are contacting them after they do so.