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8 year oldThe US president-elect said Chuck Jones had done a "terrible job" for workers at Carrier, moments after Mr Jones had criticised Mr Trump on CNN.
Mr Trump had wrongly claimed 1,100 jobs in Indiana were saved, said Mr Jones.
Earlier in the week Mr Trump attacked Boeing, hours after its boss criticised his trade policy, but he denied a link.
Shortly after Mr Trump's tweet attacking Mr Jones on Wednesday night, the union leader started receiving phone calls threatening his children, he said.
"Nothing that says they're gonna kill me, but, you know, you better keep your eye on your kids," said Mr Jones.
Keeping jobs from migrating to lower-wage countries was a central plank of Mr Trump's successful election campaign.
And he claimed a victory last week when he struck a deal with Carrier's parent company, granting them $7m in tax cuts and incentives over 10 years.
In the CNN interview that preceded the enraged tweet by the president-elect, Mr Jones disputed Mr Trump's claim on the job figures because 550 jobs were still going overseas.
Using stronger language when speaking to the Washington Post earlier this week, Mr Jones said the billionaire businessman had "lied his ass off".
On Thursday morning, the union boss admitted his choice of words was unwise but he stood by his accusation.
"Trump didn't tell the truth and I called him out," he said, saying the president-elect "overreacted".
He said that while the union was grateful to have 730 of its members keeping their jobs, Mr Trump had raised false hopes for hundreds of others when he wrongly claimed 1,100 jobs were not moving abroad.
Mr Trump's tweet sparked a back and forth on Twitter with the union, which sprang to Mr Jones's defence by saying he worked tirelessly to save "all jobs".
Mr Trump responded by tweeting that the union was to blame for jobs going abroad and it should reduce its dues.
The union fired back by saying its dues helped the union save jobs, adding the hashtag #imwithchuck
The spat is the second time this week that the president-elect has attacked an organisation that has criticised him.
On Tuesday, he threatened to cancel a huge government contract with Boeing after the chief executive made pro-trade remarks that were reported in the Chicago Tribune.
But Mr Trump said he had not seen the article in question.
The Republican president-to-be has been assembling his administration team in preparation for assuming office on 20 January.
On Wednesday there were four new additions:
Trump picks border hawk for security job
Trump picks climate sceptic Pruitt
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