News media reports, first published by the New York Times, that Trump has a plan to replace his embattled secretary of state with Pompeo, a former USA congressman who now heads the Central Intelligence Agency, drew mixed reviews from serving us officials.
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.
That Mr Tillerson has struggled to forge a comfortable relationship with Mr Trump has always been an open secret in Washington.
"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 USA diplomat who now teaches at Brown University.
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.
While acknowledging leaking the plan to journalist might be designed as an overt "signal to [Tillerson] that it is time to go", the Times reports that the "shake-up of the national security team", apparently being orchestrated by White House Chief of Staff John Kelly, "would happen around the end of the year or shortly afterward".
"He's here. Rex is here", Trump said in the Oval Office when asked if he wants Tillerson to remain in his job.
A State Department staffer told Vanity Fair he wasn't sure about the rumors, but he welcomed the plan. "I serve at the appointment of the president and I am here as long as the president feels I can be useful to achieving his objectives".
White House spokeswoman Sarah Huckabee Sanders says there are "no personnel announcements at this time".More news: Hundreds of United States massage therapists accused of sexual assault
Tillerson, 65, has spent much of his tenure trying to smooth the rough edges of Trump's unilateralist "America First" foreign policy, with limited success.
Friction between the president and the nation's top diplomat has grown increasingly public through the year.
"He remains secretary of State", she added.
If carried out, the staff changes would be the latest in a string of firings or resignations in the Trump administration including the departures of the chief of staff, national security adviser and Federal Bureau of Investigation director.
Yet divisions on key foreign policy issues emerged quickly, and Trump has repeatedly undermined Tillerson by voicing positions at odds with those the State Department was pushing.
Current and former officials said Pompeo was likely to get along better with Congress and with the White House, not least because of his conservative bent.
Tillerson joined Defense Secretary Jim Mattis in pressing a skeptical Trump not to pull the United States out of an agreement with Iran and world powers over Tehran's nuclear capabilities.
Tillerson has taken a more hawkish view than Trump on Russian Federation and tried to mediate a dispute after four Arab nations launched a boycott of Qatar.
In late September, while on a trip to Beijing, Tillerson said Washington was probing North Korea to see whether it was interested in dialogue, and had multiple direct channels of communication with Pyongyang. Trump also once tweeted that Tillerson was "wasting his time" on diplomatic efforts with North Korea.