The relationship between the president and his top diplomat has been dogged for months by reports of a rift.
Yesterday, these came to a head when several US media outlets citing White House sources predicted Tillerson's resignation.
Asked by reporters whether leaks from the White House were designed to force him out, Tillerson said: "It's laughable."
The White House did little to downplay the reports, but a State Department spokeswoman insisted Tillerson's position was safe.
Today, Tillerson himself brushed off the challenge at a photo call with Libya's Prime Minister Fayez Serraj.