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 was one of the most blatant revolving door cases in the Trump administration and a clear sign that Trump's government was of, by, and for the fossil fuel industry", said Louaillier in a statement.
"It seems like an odd way to run an administration to put a story out there that seems to imply that Tillerson is on borrowed time, but we're gonna keep him in the job for two months", said Wallace. But he now appears poised to lose an argument: over Tillerson's objections, the president is poised to recognize the disputed city of Jerusalem as the capital of Israel.
If Cotton were to take the Central Intelligence Agency job, Arkansas Gov. Asa Hutchinson, a Republican, would select someone to fill the vacant seat - and Republican officials would have a tight time frame to field a candidate who can win statewide in 2018, even though Arkansas is a deep red state.
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.
Asked about Tillerson, White House spokeswoman Sarah Sanders said the secretary of state remained in his post. Other officials say the Trump-Tillerson relationship has been souring for some time, despite Tillerson's denials.
The attempt at public shaming is just the latest such instance emanating from the Trump White House, where the President has repeatedly publicly undermined his secretary of state and publicly berated his attorney general. That is giving West Wing officials pause, a source close to the White House said.
Republican Sen. Tom Cotton's office says he's focused is on serving the people of Arkansas in the Senate.
Mr. Trump has been publicly frustrated with his top diplomat. The president was here today with the secretary of state. Trump also once tweeted that Tillerson was "wasting his time" on diplomatic efforts with North Korea.
"And I can tell you who is going to win".