Mr Tillerson could go within weeks, administration sources said today, with Mr Pompeo, a former congressman who has impressed Mr Trump at his daily intelligence briefings, being replaced at Langley, the CIA headquarters, by Tom Cotton, a hawkish Republican congressman from Arkansas.
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.
When Tillerson was tapped for the job late a year ago, many Trump critics expressed quiet relief that he'd picked a sober "adult" who could form a counterweight to the president's brasher, impulsive approach, especially on critical matters of war and peace.
State Department spokeswoman Heather Nauert said Kelly told Tillerson's chief of staff on Thursday the reports on Tillerson being replaced were not true.
"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.
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.
Kelle Louaillier, president of Corporate Accountability International, said that while ousting Tillerson, former CEO of ExxonMobil, might otherwise be a positive step, having Pompeo in charge of the U.S. State Department would be nothing to celebrate.
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.
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.
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. But Trump never signed off, and officials have struggled to focus his attention on the matter, the individual said.
In September in 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.
Mr Tillerson has also defended the Iran nuclear deal, which Mr Trump last month disavowed.
His tenure has coincided with the departure of dozens of veteran foreign policy hands, and numerous State Department's senior policymaking positions remain unfilled.