Mr. Obama offered a preview of how he plans to pitch Mr. Biden, less as a traditional stand-alone candidate than as the standard-bearer for a larger Democratic coalition bound by decency and competence. And after drawing Mr. Sanders into that fold, calling him “an American original, a man who has devoted his life to giving voice to working people’s hopes,” he concluded the video with: “Join us. Join Joe.”

The camps are still working out the details of engaging Mr. Obama in fund-raising. But David Plouffe, who remains Mr. Obama’s most trusted political adviser, has offered to pitch in, and plans to participate in several virtual Biden fund-raisers that could be a dry run for Mr. Obama’s participation, according to people briefed on the plans.

Mr. Biden’s emergence as the Democrats’ presumptive nominee relatively early in the political calendar is unwelcome news to Mr. Trump, his bluster notwithstanding, several of the president’s advisers said. Last Thursday, after trying to goad an anti-Biden revolt among Sanders supporters, the president suggested dark motives for Mr. Obama’s hesitancy in endorsing Mr. Biden.

“You know what? I’ll tell you, it does amaze me that President Obama hasn’t supported Sleepy Joe,” Mr. Trump said at a White House coronavirus briefing, in between questions about his administration’s response to the crisis. “It just hasn’t happened. When is it going to happen? When is it going to happen? Why isn’t he? He knows something that you don’t know, that I think I know, but you don’t know. So it’ll be interesting.”

That claim was Trumpian misdirection. Mr. Obama timed his video to follow Mr. Sanders’s endorsement on Monday.

But the Biden-Obama relationship, which deepened from a congenial partnership into a real friendship in 2015, when the president consoled Mr. Biden during his son Beau’s illness and death, is not without complications.

Mr. Biden is grateful for Mr. Obama’s friendship but increasingly proud of his historic comeback. When news reports surfaced that Mr. Obama had called to congratulate Mr. Biden on his victory in South Carolina, the candidate made it clear to his staff that while his connection to Mr. Obama played a role in delivering African-American voters, Mr. Obama “had not lifted a finger” on his behalf, according to a senior Democrat with knowledge of his remarks.