The reservations of union members could be a bigger problem for Mr. Biden than they were for Hillary Clinton during her 2016 Democratic race against Mr. Sanders. Some large unions, including the American Federation of Teachers, endorsed Mrs. Clinton, though many members later supported Mr. Sanders.

In the current cycle, many of these unions have skipped an early endorsement, making it easier for individual members and in some cases locals to support their own candidates. The teachers’ union in Los Angeles has endorsed Mr. Sanders, as has the Ottumwa local of the United Food and Commercial Workers, whose 1.3-million-member international endorsed Mrs. Clinton before the 2016 Iowa caucuses. A large Pennsylvania local of the food workers’ union has endorsed Mr. Biden.

While the Labor Department recently reported that union membership last year fell to a record low — 10.3 percent of the work force — labor endorsements can still be critical because of the role of unions in educating members about candidates and canvassing for them on the ground.

Mr. Laursen, the U.A.W. local leader in Ottumwa, estimates that more than half his members — who are primarily workers at a John Deere plant — backed Mr. Trump in 2016. But he says many of those who oppose the president’s re-election are supporting Mr. Sanders over Mr. Biden.

And the skepticism toward Mr. Biden among union voters may be even more pronounced in the less white, less male parts of the labor force.

Nicole McCormick, a West Virginia music teacher who helped organize a statewide walkout that made national headlines in 2018, said she worried that Mr. Biden wasn’t “willing to push for the things that we as Americans look at as radical, but the rest of the world looks at and is like, ‘We did that 50 years ago.’” She cited expanded access to unions, universal health care and paid parental leave as examples.