Trump tweeted Wednesday evening: "Chuck Jones, who is President of United Steelworkers 1999, has done a terrible job representing workers."
That local union branch represents workers at Carrier's Indianapolis plant.
In a second tweet, Trump suggested Jones should "Spend more time working - less time talking" and the union should "Reduce dues."
About 30 minutes after Trump tweeted about Jones the union leader started getting harassing phone calls, he told MSNBC on Wednesday.
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He told the cable news outlet he wasn't sure how the callers found his number.
"Nothing that says they're gonna kill me, but, you know, you better keep your eye on your kids," Jones told MSNBC.
"I've been doing this job for 30 years, and I've heard everything from people who want to burn my house down or shoot me. So I take it with a grain of salt and I don't put a lot of faith in that, and I'm not concerned about it and I'm not getting anybody involved. I can deal with people that make stupid statements and move on."
He accused Trump and Vice President-elect Mike Pence of staging "a dog and pony show" around the deal.