Story highlights "I would say the ice continues to thin," the source said

Multiple senior Republican sources say they don't expect Tillerson to last past January

(CNN) Secretary of State Rex Tillerson's future in the Trump administration is looking increasingly uncertain, an administration source tells CNN.

"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.

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