Tillerson speaks on behalf of the president and the country.
Mr Tillerson, who is scheduled to have lunch with Mr Trump and Defence Secretary Jim Mattis on Tuesday, held a special press conference last week to deny considering quitting.
Mr Trump also took to Twitter to deny the story last week, writing: "Rex Tillerson never threatened to resign".
While the two men have publicly professed a positive relationship with one another, Trump and Tillerson have appeared to clash at times, most recently on the issue of North Korea.
"No, I didn't undercut anybody".
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Tillerson's departure would be a major blow to those hoping to temper Trump and stop what Corker described as "the path to World War III". "I think I'm actually strengthening authority".
Later, the State Department said Tillerson never used "that type of language". Regardless of who would win that competition - Trump implied he win - it would do nothing to answer the question of who is smarter. Trump tweeted early Tuesday morning in the US.
Tom Price, Trump's first Health and Human Services secretary, resigned in late September over a private plane scandal, and since becoming President, Trump has dismissed several top communication aides, a chief of staff and a chief strategist.
It's hard to know which is more devastating for the nation - a president so utterly lacking in intelligence and intellectual curiosity that he must be "handled" by his staff to avoid a deadly world catastrophe, or that the fact that he is so inept that foreign powers "play dumb" in an attempt to make global relations accessible for him.