Addressing the striking U.A.W. workers in Kansas City on Sunday afternoon, Mr. Biden wore a red T-shirt in solidarity. He referenced the bailout of General Motors during the Obama administration, telling the crowd, “We didn’t bail out GM, U.A.W. bailed out GM!” He went on to lament the high pay of executives at the company and the lack of equitable benefits for workers, and he encouraged the striking workers, acknowledging, “you’re making a hell of a sacrifice.”

“There’s only one reason we have a middle class, and it’s spelled ‘U-N-I-O-N,’” he said. The crowd cheered.

Auto-worker support for the Democratic Party has eroded slightly in recent years, and about 30 percent of the union’s rank and file were estimated to have voted for Mr. Trump in 2016, slightly more than the share who voted for the two previous Republican presidential nominees.

The departure of those voters from the Democratic Party, even as the union had officially endorsed Hillary Clinton, was particularly critical to Mr. Trump in Michigan, where he won by the thinnest of margins — less than 11,000 votes — to capture the state’s 16 electoral votes.

Nearly three years later, it is unclear how many of those union workers continue to back the president. The economy has generally remained strong here, with unemployment near a 20-year low.

Representative Debbie Dingell, a Democrat who represents a district just outside of Detroit that is home to several auto manufacturing facilities, said she felt that Mr. Trump’s support among autoworkers had held roughly steady.

But she said she believed that a prolonged strike could move the president’s numbers among such workers in one direction or the other, depending on whether they viewed him as trying to help or hurt them.