And according to a dozen Republican donors and strategists, as well as former elected officials who can speak free from concerns about retribution from Mr. Trump, there is some evidence that Mr. Biden can connect with these voters.

“Oh yeah, he definitely does,” said former Gov. John Kasich of Ohio, a 2016 Republican presidential candidate, pointing to Mr. Biden’s ability to speak to blue-collar voters. “He is a guy who can do it probably better than any other Democrat.”

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Chuck Hagel, a Republican who served as defense secretary in the Obama administration, said he wrote in Mr. Biden’s name on the ballot instead of voting for Mr. Trump or Hillary Clinton in 2016. This year, he said, he favors Mr. Biden again, and would not back Mr. Trump — or Ms. Warren, or Senator Bernie Sanders of Vermont — in the general election. It’s a view shared by other Republicans, Mr. Hagel added.

“They’ve said to me, ‘If Biden is the nominee, I will vote for Biden, I will not vote for any of the other Democrats,’” recalled Mr. Hagel, a former Republican senator from Nebraska, citing Mr. Biden’s experience and his empathy. “I don’t know how big or deep or wide that is in this country, but I hear it.”

At Biden campaign events, prominent supporters often instruct attendees to think in pragmatic terms, a stark departure from the fiery populist messages that animate events held by Ms. Warren and Mr. Sanders.

“It’s not going to be you, it’s not going to be me, it’s not going to be the party faithful that turn this election — it’s going to be independents and moderate Republicans,” Chris Louscher, a Democratic activist, told a crowd at a Biden event in Algona, Iowa, this month, thinking of her brother-in-law in Florida. “I have a lot of family, a lot of friends, I have people in this audience today that are Republicans. They will vote for Joe Biden and that’s how he wins this next election.”

In recent days, Mr. Biden has been sharpening the contrast between himself and his Democratic rivals, using an addition to his stump speech to swipe at those who suggest that he is “naïve” for wanting to cooperate with Republicans — the very Republicans, he acknowledges, who are attacking him, his family and his “only living son.”