In an interview published Tuesday by Forbes, Trump said that he didn't believe Tillerson called him a "moron", before expressing confidence that he would have a higher IQ score than the man he made the country's top diplomat.
A senior State Department official says Friday's meeting was last-minute and was related to the Middle East. "And I can tell you who is going to win", Trump said.
ITK reached out to the geniuses at Mensa - which describes itself as "an worldwide society whose only qualification for membership is a score in the top 2 percent of the general population on a standardized intelligence test" - and asked if they'd be willing to take Trump's suggestion.
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"American Mensa would be happy to hold a testing session for President Trump and Secretary Tillerson", spokesman Charles Brown, said. "But if he did that, I guess we'll have to compare IQ tests".
But, despite trashing his secretary of state's traditional approach to diplomacy ("we can talk to them") with Twitter-threats of nuclear annihilation, Trump insists his style is bolstering the White House team.
Meanwhile, Mr Trump has repeatedly undercut Mr Tillerson's statements on some of America's most sensitive national security challenges, including Iran and North Korea, calling it as waste of time. "I'm not undermining. I think I'm actually strengthening authority".