Mr Tillerson's departure would end a troubled tenure for the former Exxon Mobile chief executive, who has found himself increasingly at odds with Mr Trump over policy challenges such as North Korea and under fire for his planned cuts at the State Department.
The White House is considering a major shakeup of President Donald Trump's national security team in the coming weeks, according to a report by the New York Times's Peter Baker and Maggie Haberman.
But White House spokeswoman Sarah Huckabee Sanders didn't deny it.
Trump has soured on Tillerson mostly because of the "moron" report, his less confrontational approach on North Korea and differences over the Qatar crisis, one senior US official said.
Under the plan, Republican Senator Tom Cotton would be tapped to replace Mr Pompeo at the Central Intelligence Agency, the New York Times reported, citing senior officials.
Baker and Haberman write that it is "not immediately clear whether Mr. Trump has given final approval to the plan", but that Chief of Staff John Kelly has drawn it up and widely circulated it.
The next day, Trump appeared to dismiss those efforts in a tweet, telling Tillerson he was "wasting his time" trying to negotiate with North Korea.More news: Suspect pleads not guilty to deadly NY truck attack
Tillerson's top priority as secretary has been his sweeping overhaul of the State Department, a "redesign" that has been lambasted by lawmakers from both parties and that the State Department concedes has hurt morale among diplomats. Too much churn could fuel the perception of chaos in the Trump White House - perhaps one reason he has yet to pull the trigger.
Friction between the president and the nation's top diplomat has grown increasingly public through the year.
Wallace questioned why the reported plan calls for Tillerson to stay until late January, adding the secretary is scheduled to make a trip to Europe next week to meet with key USA allies. But Trump never signed off, and officials have struggled to focus his attention on the matter, the individual said.
While some intelligence officers say Pompeo tends to tell the president what he wants to hear rather than giving him their assessments, others say they have been impressed by his intellect, his willingness to listen and his advocacy of more robust covert operations.
Now, CBS News has confirmed the White House has a plan to force Tillerson out.
Tillerson has taken a more hawkish view than Trump on Russian Federation and tried to mediate a dispute among key U.S. Mideast allies, after four Arab nations launched a boycott of Qatar, alleging Islamist extremist ties.
Tillerson successfully convinced Mr. Trump to keep the U.S.in the Iran nuclear deal.
His tenure has coincided with the departure of dozens of veteran foreign policy hands, and numerous State Department's senior policymaking positions remain unfilled.