Chuck Jones, president of United Steelworkers Local 1999, which represents workers at United Technologies Corp's Carrier plant in Indianapolis, in an interview on CNN criticised Trump for at least the second time in two days over the deal.
Trump, in a Twitter post on Wednesday, offered a sharp reply: "Chuck Jones, who is President of United Steelworkers Local 1999, has done a terrible job representing workers. No wonder companies flee country!"
On Tuesday, Jones told The Washington Post that when Trump spoke at the Carrier factory last week, "He got up there and, for whatever reason, lied his ass off."
Indiana state officials have agreed to give Carrier $7 million in tax breaks to encourage it to keep about 1,100 jobs in the state, although about 300 of the jobs are in the company's headquarters and were not scheduled to go to Mexico.
Only 800 of the positions are union factory jobs, and Carrier still plans to send an estimated 1,300 jobs to Mexico.
Some employees are disappointed, Jones said on CNN on Wednesday. "We had a lot of our members, when the word was coming out of 1,100, they thought that they would have a job," he said.
In more remarks on CNN later on Wednesday, Jones responded to Trump's Twitter post. "That's pretty low-down, low life," he said. "He doesn't like to be challenged, and I challenged him."
Trump, who made tough talk on trade a centrepiece of his election campaign, had vowed to impose hefty tariffs if Carrier did not reverse course on shifting jobs to Mexico.
Trump posted another tweet after Jones' later remarks on CNN. "If United Steelworkers 1999 was any good, they would have kept those jobs in Indiana," Trump wrote.
