Mr. Trump’s intervention in a local labor-management issue was highly unusual for an incoming or sitting president, according to several labor historians. But the episode, which began last week with Mr. Trump holding a triumphant press conference at Carrier and devolved into a bitter Twitter exchange after United Steelworkers local president Chuck Jones accused Mr. Trump of exaggerating the number of jobs saved, has made union leaders wary.
Labor leaders came to the defense of Mr. Jones who said he received telephone calls threatening him and his family after the president-elect’s tweets.
“Taking him on is not like taking on Hillary Clinton. This is an everyday guy,” said Greg Junemann, president of the International Federation of Professional & Technical Engineers, which represents about 95,000 members, including workers at Boeing Co. Nevertheless, Mr. Junemann said, “I do commend Trump for saving the jobs.”
During the presidential campaign, major labor groups and leaders backed Mrs. Clinton over the Republican even as Mr. Trump held his own among union members. Exit polls showed that 43% of voters in union households went for Mr. Trump, just 8 percentage points behind Mrs. Clinton.
Some labor leaders said they are hopeful that unions can find common ground with Mr. Trump on issues like trade and creating jobs for infrastructure projects. “If you take him at his word opposing these rotten trade deals, then that’s an area where we can have a lot of unity,” said Mark Dimondstein, president of the American Postal Workers Union, which represents about 200,000 workers.
But the Carrier episode combined with Mr. Trump’s picks for his cabinet, including his labor secretary pick, have drawn sharp criticism from labor leaders. On Friday, several union officials declined to comment on the incoming administration, saying they were waiting for the Carrier spat to settle down.
Several labor historians said it would be unprecedented if Mr. Trump were to continue to take an activist role with companies and unions, especially at the local level.
In the past, presidents typically only stepped in during strikes that would disrupt the economy. President Ronald Reagan fired 11,000 air-traffic controllers in 1981 after they engaged in an illegal strike. But for other examples, historians had to reach back to the 1940s when massive strikes in the coal, steel or other industries could derail interstate commerce.
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“It was not unusual for [President Harry] Truman to comment on strikes,” said Joshua Freeman, a labor historian at City University of New York. “The context was different. These were strikes and these were situations involving tens or hundreds of thousands of workers disrupting the whole national economy.”
Even in those earlier cases, however, presidents typically worked within a framework established by federal labor law to help resolve conflicts, he said. Mr. Trump, by contrast, is far more freewheeling and unpredictable.
Even so, some experts said they believe the Carrier dust-up is a harbinger of future friction between Mr. Trump and labor.
Nelson Lichtenstein, a labor historian at the University of California, Santa Barbara, predicted a rocky four years for unions under Mr. Trump. “I think it will be at best trench warfare, and probably a series of explosive conflicts in which the Trump administration is the aggressor,” he said.
Mr. Trump has said that by stepping in he kept 1,100 Carrier jobs from moving to Mexico. But Mr. Jones, the president of the United Steelworkers local that represents workers at the Carrier plant, said that included 350 engineering jobs that the company never intended to relocate. Mr. Jones said Friday that he wanted to set the record straight, in part because the 1,100 number gave some workers false hope that their jobs would be saved. The union said 550 jobs would still be lost at the plant.
After Mr. Jones’s initial statement, the president-elect blasted the labor leader on Twitter, saying he had done “a terrible job representing workers. No wonder companies flee country!”
In a follow-up tweet, Mr. Trump criticized the union and said it should reduce membership dues and “spend more time working-less time talking.”
“If United Steelworkers 1999 was any good, they would have kept those jobs in Indiana,” Mr. Trump wrote.
Various unions came to Mr. Jones’s defense, including on Twitter, where Richard Trumka, head of the AFL-CIO, said, “#ImWithChuck because an attack on him is an attack on all working people.” Similarly, Mr. Dimondstein of the postal-workers union said the president-elect’s swipes against Mr. Jones represented “a very ominous sign for this country.”
A spokesman for Mr. Trump couldn’t be reached to comment.
Reached on Friday, Mr. Jones said he hadn’t heard anything from Mr. Trump since the tweets but would be willing to meet with him.
“I’ll give the guy the benefit of the doubt, even though I wasn’t a Trump supporter,” Mr. Jones said. “If he’s interested in truly trying to save these jobs I would be willing and able to sit down and work with him.”
Some current and former union members also said they want to give Mr. Trump a chance to make good on his promise for jobs, especially in the Rust Belt.
Buzzy Byron, a 62-year-old contractor and former steelworker union member in Monessen, Pa., said he decided “at the last minute” to vote for Mrs. Clinton but now wants Mr. Trump to succeed.
“I dusted off my hard hat, and I’m ready to go to work,” Mr. Byron said.
Write to Kris Maher at kris.maher@wsj.com
