Asked by a reporter whether he wanted Tillerson to stay on the job, Trump was coy, merely pointing out that Tillerson was in fact in the building.
The story originated in the New York Times, and officials are saying that it actually got there directly from the White House, as a way of White House officials "sending a message" to Tillerson. Both Republican Sen. Tom Cotton of Arkansas and retired Navy Vice Adm. Robert Harward are under consideration for that post.
Sen. Tom Cotton (R-Ark.) would then be tapped to head up the Central Intelligence Agency, according to reports.
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.
Tillerson, the former chief executive of Exxon Mobil, has had a testy relationship with President Trump at times.
Mr Trump surprised many observers when he tweeted that Mr Tillerson was "wasting his time" pursuing contacts with North Korea, and Mr Tillerson was quoted as having said that the president was a "moron", though he denied having said it.
Relations between President Trump and the senior member of his Cabinet - the former CEO of Exxon Mobil - have been strained nearly from the start. And the plan is not just being considered at the staff level, but by the President himself, the source said. "Not only will the Koch Brothers now have a direct line to the State Department, Pompeo will lead US foreign policy down the same dark, bigoted and Islamophobic path of Donald Trump's Twitterfeed". "When the president loses confidence in someone, they will no longer serve in the capacity that they're in", she said. Bill to save cyber diplomacy office advances America's decision on North Korea hinges on Trump's success in Asia MORE.
Tillerson joined Defense Secretary Jim Mattis in pressing Trump not to pull the United States out of an agreement with Iran and world powers over Tehran's nuclear capabilities.
When Tillerson in June called on Arab nations to ease their blockade on Qatar, Trump emerged in the Rose Garden hours later to lambaste Qatar for funding terrorism. But Trump never signed off, and officials have struggled to focus his attention on the matter, the individual said.
As a member of Congress, she noted, Pompeo represented Wichita, Kansas, where Koch Industries is based, has been called the "Koch Brothers' favorite congressman".
Several administration officials said that Pompeo has said previously he's open to the job.
The timing of Tillerson's expected departure remains uncertain, but multiple officials said the shake-up could come at the end of the year or early next.