Last week, after NBC News reported that Tillerson, increasingly angry with Trump and other administration figures, referred to Trump as "a moron", the president dismissed the story as "phony".
But in an interview with Forbes magazine published on Tuesday, Trump said that if the NBC report was true, he'd compare IQ test results with Tillerson and come out the smartest.
"I think it's fake news," Trump said in the interview. "But if he did that, I guess we'll have to compare IQ tests. And I can tell you who is going to win."
Tillerson last week gave an unscheduled press conference to address the story, praising his boss as "smart" and affirming his "commitment to the president and the success of our country". Notably, he did not specifically deny calling Trump a moron.
The secretary of state left the job of disputing the story to State Department spokeswoman Heather Nauert, who told reporters, "The secretary does not use that type of language."
Later that day, Trump said that he appreciated Tillerson showering him with praise. Despite rumblings that Tillerson may be on his way out, the president said he had "total confidence in Rex".
Trump on Tuesday is scheduled to have lunch with Tillerson and Defense Secretary Jim Mattis.