Mr Tillerson is due to embark on a major European trip on Monday but preparations have been clouded by anonymous White House leaks suggesting he was to be replaced within weeks.
But Mr Trump took to Twitter after lunch with Mr Tillerson to declare the speculation "FAKE NEWS!" -- just hours after the secretary had brushed off the leaks as "laughable."
Media reports on Thursday, first published by the New York Times, cited US officials as saying that Trump has a plan to replace his embattled secretary of state with Pompeo, a former US congressman who now heads the Central Intelligence Agency.
Asked if some at the White House wanted him to resign, Tillerson told reporters: "It's laughable ... laughable."
His comments came as he posed for pictures with Libyan Prime Minister Fayez Seraj of the UN-backed government in Tripoli.
Tillerson's relationship with Trump has been strained by the top US diplomat's softer line on North Korea and other policy differences, as well as by reports in October that he had called the president a "moron".
Tillerson has not directly addressed whether he made the comment, though his spokeswoman denied it.
Asked on Thursday whether he wanted Tillerson to remain in his job, Trump sidestepped the question, telling reporters at the White House: "He's here. Rex is here."
