Mr. Biden’s lackluster debate performances and alarmingly low cash flow — he has less than $9 million on hand, not even half of some of his rivals — has fueled the Democratic disquiet. But if the causes of the concern are plain to see, what exactly can be done about it is less clear.

And even some of those being wooed acknowledge that it can be hard to discern between people just being nice and those who genuinely want them in the race.

Mrs. Clinton and Mr. Bloomberg have both told people privately in recent weeks that if they thought they could win, they would consider entering the primary — but that they were skeptical there would be an opening, according to Democrats who have spoken with them.

Former Secretary of State John F. Kerry, who associates say has wondered aloud about whether he should have run and has found it hard to watch Mr. Biden’s missteps, has also been urged to get in. But he still thinks the former vice president, who was once his longtime Senate colleague, is the party’s best nominee.