WASHINGTON -- In 10 months as America's chief diplomat, Secretary of State Rex Tillerson has found no world leader a greater challenge than the President of the United States.

Relations between President Trump and the senior member of his Cabinet -- the former CEO of Exxon Mobil -- have been strained almost from the start.

Now, CBS News has confirmed the White House has a plan to force Tillerson out.

Mr. Trump on Thursday refused to answer questions about a White House plan to replace Tillerson after less than a year on the job.

Mr. Trump has been publicly frustrated with his top diplomat.

Secretary of State Rex Tillerson and President Trump. CBS News

"We disagree on a couple of things," Mr. Trump has said. "Sometimes I'd like him to be a little bit tougher."

That comment came just three days after Tillerson held an extraordinary news conference last month to shoot down reports that he had called the president a moron.

"There has never been a consideration in my mind to leave," Tillerson said. "I serve at the appointment of the president and I am here as long as the president feels I can be useful to achieving his objectives."

The two have clashed on policy issues, including North Korea. Shortly after Tillerson announced that the U.S. was in diplomatic contact with Pyongyang, Mr. Trump tweeted that "he is wasting his time."

"No, we don't agree on everything," Tillerson told John Dickerson on "Face the Nation" in October. "Yes, sometimes, he changes his mind."

Tillerson successfully convinced Mr. Trump to keep the U.S. in the Iran nuclear deal. 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.

One of Tillerson's closest allies, Senate Foreign Relations Committee Chairman Bob Corker, said Mr. Trump has effectively neutered Tillerson.

Corker "said the president quote has castrated you before the world stage," CNN's Jake Tapper said to Tillerson.

"I checked. I'm fully in tact," Tillerson said.

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.

Corker on Thursday defended Tillerson.

"It's been evident to me that for some time, somebody is, you know, seeking to undermine his presence here," Corker said. "I don't know who that is."

The White House insists no change is imminent. The favored candidate to replace Tillerson is current CIA Director Mike Pompeo, which would leave a vacancy at the CIA. Both Republican Sen. Tom Cotton of Arkansas and retired Navy Vice Adm. Robert Harward are under consideration for that post.