His retaliation came in a Forbes magazine interview published yesterday.
NBC News broke the story last week that Tillerson openly called the president a moron, leading to a weird press conference in which Tillerson said Trump "is smart" and announced that he wasn't quitting.
In this October 5, 2017, photo, Secretary of State Rex Tillerson arrives for the first meeting of the National Space Council in Chantilly, Va. President Donald Trump challenged Tillerson to "compare IQ tests", delivering a sharp-edged ribbing that threw a bright spotlight on his seemingly shaky relationship with his top diplomat.
Both Trump and Tillerson graduated from university with Bachelors of Science, majoring in economics and civil engineering respectively.
"American Mensa would be happy to hold a testing session for President Trump and Secretary Tillerson", spokesman Charles Brown, said.
President Donald Trump stands behind and in front of members of a Marine honor guard as he greets Canadian Prime Minister Justin Trudeau and Sophie Gregoire Trudeau as they arrive at the White House in Washington, Wednesday, Oct. 11, 2017.
"I'm not undermining. I think I'm actually strengthening authority", Mr Trump said.
"I think it's fake news, but if he did that, I guess we'll have to compare IQ tests".
Tension has been simmering between the pair for some time over North Korea, with Trump's aggressive rhetoric at odds with Tillerson's more measured approach.
He produced two sets of IQ test results, one for Tillerson and one for Trump. "And I can tell you who is going to win", Trump told Forbes reporter Randall Lane. He points to Trump's derision of "liddle Bob Corker", the senator who voiced the fear that Trump could be taking the country towards "World War III".