"When the president loses confidence in someone, they will no longer serve here", White House spokeswoman Sarah Sanders said when asked about reports of a staff shake up.
Tillerson is the former Exxon Mobil CEO who has had tensions with Trump in several foreign policy areas, as well as overwhelmingly negative publicity over plans to reorganize the State Department.
Tillerson's likely ouster loomed awkwardly over an Oval Office meeting Thursday between Trump and the visiting Bahraini crown prince.
Tillerson answered that "the president speaks for himself". The White House has a plan to force out Secretary of State Rex Tillerson and replace him with Central Intelligence Agency chief Mike Pompeo within weeks, the New York Times reported.
Tillerson's aloof management style and cost-saving plan to restructure the State Department, including slashing around 2,000 positions, has not helped his popularity.
In 10 months as America's chief diplomat, Secretary of State Rex Tillerson has found no world leader a greater challenge than the President of the United States.
This combination of pictures created on November 30, 2017 shows US Secretary of State Rex Tillerson (left) and CIA Director Mike Pompeo in Washington, DC. Tom Cotton, R-Arkansas, could present a wrinkle in the strategy.
A source familiar with Tillerson's thinking said the secretary of state's original plan when he took the job was to leave in February.
The White House insists no change is imminent.
"There has been a kind of Cold War going on between these two", Wallace noted, adding there has seemed to be a rift between Trump and Tillerson on handling North Korea.
"If the president undercuts what he is trying to achieve diplomatically, convinces people that whatever you agree to with the secretary of state will be overturned by the president ... then he is basically neutered", said Richard Boucher, a former top US diplomat who now teaches at Brown University.