Since announcing his candidacy last week Mr. Biden has repeatedly invoked his decades-long bond with organized labor. In his first campaign speech on Monday, at a Teamsters building in Pittsburgh, Mr. Biden thanked a litany of unions, from teachers to steelworkers to carpenters, and declared, “I make no apologies: I am a union man.’’

Mr. Trump’s campaign indicated that they intend to hire an aide focused on union outreach, and the president himself has wooed some moderate national labor leaders. He has twice held events at local facilities of the International Union of Operating Engineers, and last month in Texas he made the case to Jim Callahan, the union’s president, for why they should back his re-election, according to a Trump official briefed on the conversation.

But any slippage he suffers with this blue-collar constituency could prove politically fatal next year: He was elected by only the narrowest of margins in a handful of states in the industrial Midwest.

What worries the president’s aides about Mr. Biden, according to people familiar with their thinking, is that his moderate profile and deep ties to labor could win back many of the voters who twice backed President Barack Obama before spurning Hillary Clinton.

“Start right here in Pennsylvania and work your way up through Ohio, Michigan — that’s going to be the battleground and I think they come home,” Harold Schaitberger, the head of the pro-Biden International Association of Fire Fighters, said earlier this week in Pittsburgh.