In a tweet, a White House reporter quoted Mr Trump saying: “Arnold Schwarzenegger... You know what? He died... I was there.”
But just one hour later Mr Schwarzenegger, who is not dead, quoted the tweet and added: “I’m still here. Want to compare tax returns, @realDonaldTrump?”
The journalist clarified the president had been referring to The Terminator star’s TV ratings when he said the 71-year-old had “died”.
NBC cancelled The Apprentice in 2015 after Mr Trump had disparaged immigrants at the start of his presidential campaign.
But in 2017 the former governor of California took over the boardroom hotseat when the network tried to relaunch the show.
However, Mr Schwarzenegger only filmed one season of The Apprentice and said the reboot failed because of its continued association with Mr Trump.
The feud between the two septuagenarians was reignited when Mr Trump attacked his rival’s performance on the reality TV programme this week.
“I gave them a top show when they were dying on NBC. But they don’t like me too much. They wanted a big extension,” he said.
“They used Arnold Schwarzenegger instead. Big movie star. You know what? He died. He died. I was there 12 years, 14 seasons and then they pick a movie actor and he dies on us.”
But Mr Schwarzenegger has now hit back by suggesting the president should release his tax returns.
Mr Trump broke decades-long convention by refusing to publish his financial statements during the election campaign.
In May, The New York Times obtained some details of the real estate tycoon’s taxes, which revealed he lost so much money during the 1980s and 1990s he paid no income tax at all for eight years.
The newspaper reported Mr Trump lost more money than almost any other taxpayer during this period, a total of $1.17bn (£934m).
Democrats in Congress are also engaged in a legal tussle in an effort to force the White House to release the last six years of tax returns – a period not covered by The New York Times leak.
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