Mr. Sanders, speaking after last week’s losses in Michigan and three other states, telegraphed his intention to press Mr. Biden hard on health care, climate change and income inequality. Yet the actions of both men in the aftermath of Tuesday’s primaries — Mr. Biden set out a welcome mat for his rival rather than pressuring him to quit, and Mr. Sanders outlined tough terms for an eventual détente — also shed light on a personal relationship that has remained sturdy, amicable and functional, a far cry from the acrimony that defined Mr. Sanders’s relationship with Mrs. Clinton after their bitter duel four years ago.

This no-frills personal connection, aides to both men say, could become an important factor in quickly uniting the party to confront President Trump, despite the wide policy gulf between the moderate Mr. Biden and the progressive Mr. Sanders.

“I think both men know where this is likely going, and they both know how to approach one another,” said David Axelrod, a longtime adviser to President Barack Obama who worked closely with Mr. Biden in the White House. “Sanders, it seems to me, is a guy who wants to land the plane, and is asking Biden to show him some lights from the ground. There’s nothing personal standing in the way of them getting together.”