"I would say the ice continues to thin," the source said. "The question is when does it actually break."
During a Sunday interview with CNN's Jake Tapper on "State of the Union," Tillerson maintained that he's fully committed to the job. He said he wouldn't dignify confirming or denying reports indicating that he called President Donald Trump a "moron" in a private conversation
with fellow US officials following a meeting with the President at the Pentagon. While Tillerson has repeatedly declined to flatly deny using the insult to describe Trump, Heather Nauert, a State Department spokeswoman, has denied Tillerson ever called the President a "moron."
When the "moron" story became public, Health & Human Services Secretary Tom Price had just resigned
over his private jet use on the federal government's dime. Senior Republican sources say Tillerson wasn't axed then because White House chief of staff John Kelly didn't want to reinforce the "chaos" theme surrounding the high turnover rate within the administration.
Multiple senior Republican sources say they don't expect Tillerson to last past January. Some names being floated to replace Tillerson include US Ambassador to the UN Nikki Haley and CIA Director Mike Pompeo.
While Haley has long been mentioned as a contender, some members of the administration are lobbying against her and don't want her to have such a highly visible job, the GOP sources said.
Pompeo, who briefs the President nearly every day, may be the leading candidate for State, the senior Republican sources say. And Sen. Tom Cotton, R-Arkansas, who garnered support
from Trump for his immigration bill, has also been mentioned as a possible replacement for Pompeo at the CIA were he to succeed Tillerson at State, the sources said. Axios first reported
the possibility of Cotton taking over at CIA.
Tillerson has also faced critics within the Republican Party, namely, outgoing Tennessee Sen. Bob Corker, who told The Washington Post last week the President "cannot publicly castrate your own secretary of state without giving [himself] that binary choice."
Corker referenced public statements from the President -- including a recent tweet regarding North Korea in which he said Tillerson was "wasting his time" on diplomatic efforts with the US foe -- suggesting they undermined the secretary's efforts.
Behind the scenes, the senior Republican sources say Trump and Tillerson strongly dislike each other -- especially following the "moron" incident.
But Tillerson and Trump both say they get along just fine. Trump has said he has confidence
in the secretary, and Tillerson has repeatedly stated that he is committed to executing the President's objectives.