"And I can tell you who is going to win," Trump said to Forbes magazine.
Trump's tense relationship with Tillerson burst into public view last week. An NBC News story claimed Vice President Mike Pence had to talk Tillerson out of resigning this summer, and that Tillerson had called Trump a "moron."
Tillerson said he never considered resigning, though he didn't directly address the reported insult. His spokeswoman later said he never used such language.
Trump and Tillerson are scheduled to have lunch Tuesday at the White House with Defense Secretary Jim Mattis.
Trump has at times appeared to undercut Tillerson's message on some of America's most sensitive national security challenges, including Iran and North Korea. Tillerson also has publicly complained about the White House blocking him from making key appointments.
Still, last week Trump told reporters he has "total confidence" in his secretary of state.
In the Forbes interview, done Friday, Trump responded to criticism that he's undermined his secretary of state through his often provocative tweets that have interfered directly with ongoing diplomatic efforts.
"I'm not undermining," Trump told Forbes. "I think I'm actually strengthening authority."
As for Tillerson's reported "moron" comment, the president said, "I think it's fake news. But if he did that, I guess we'll have to compare IQ tests. And I can tell you who is going to win."