For Mr. Biden, who has antagonized Mr. Sanders as a non-Democrat out to diminish the presidency of Barack Obama, even a firm insistence that he understands what is driving Sanders voters will be viewed skeptically.

But despite Mr. Biden’s success during the last two weeks of primaries, he will need to make gains with some key Sanders constituencies — not only younger liberals but also Latinos across several age brackets — to build the most formidable coalition possible in the fall. And he will almost certainly need Mr. Sanders’s help to get there.

Some veterans of Mrs. Clinton’s campaign would advise Mr. Biden not to expect much. Though Mr. Sanders officially endorsed Mrs. Clinton in 2016 and rallied for her, her aides generally viewed the endorsement as somewhere between dutiful and inadequate — especially when weighed against the months of negative attacks from Mr. Sanders and his supporters depicting her as a corporate tool well after she appeared to have the nomination in hand.

There is some reason to believe that this election might be different. Mr. Sanders appears to harbor much less visceral disdain for his opponent than he did in 2016, when he and his aides seethed constantly at the perceived coronation of Mrs. Clinton. Mr. Sanders has never attacked Mr. Biden’s character or his personal ethics, even as a good number of his political allies have done so repeatedly.

“It needs to be a two-way dance,” said Brian Fallon, Mrs. Clinton’s national press secretary four years ago, noting that even a progressive like Senator Elizabeth Warren had been found wanting by Sanders fans. “What the courting looks like, what will actually satisfy large chunks of his movement, is a mystery. So much of that movement could not even find Warren as an acceptable alternative. So, what the heck can Joe Biden do?”

Mr. Fallon did have a few ideas: namely, some policy drift toward Mr. Sanders’s platform on climate or health care issues from the Biden camp if Mr. Sanders leaves the race.