But the deliberate, slow-talking oil executive has little personal chemistry with the quick-talking, impulsive Mr. Trump. Mr. Tillerson has avoided expressing his pique to the president. But aides and Trump associates who have been in the room with them said Mr. Tillerson’s body language, eye rolling and terse expressions left little doubt that he disapproves of Mr. Trump’s approach.
Mr. Trump, they said, has noticed how Mr. Tillerson slouches in his presence, particularly when he disagrees with a decision. When overruled, Mr. Tillerson often says, “It’s your deal,” to the president’s irritation, according to two former administration officials.
Mr. Tillerson felt compelled on Wednesday to address the internal schism after NBC News reported that he had been prepared to step down this summer until he was talked out of it and that after a meeting of national security officials at the Pentagon he had derided Mr. Trump as a “moron.” Vice President Mike Pence at one point counseled Mr. Tillerson on how to ease tension with the president, according to the report.
“There’s never been a consideration in my mind to leave,” Mr. Tillerson told reporters. “I serve at the appointment of the president, and I am here for as long as the president feels I can be useful to achieving his objectives.”
Asked directly if he had called Mr. Trump a “moron,” Mr. Tillerson would not say. “I’m not going to deal with petty stuff like that,” he said.
When that left the impression that it was true, a State Department spokeswoman, Heather Nauert, appeared before reporters hours later to deny it on his behalf. “He did not say that,” she said.
The spectacle left the State Department paralyzed. In the moments leading up to Mr. Tillerson’s statement, hallways in the department’s cavernous headquarters were nearly deserted as diplomats were glued to televisions wondering whether he would join the parade of top Trump administration officials who have resigned.
Indeed, the news conference was the latest rupture in an administration consumed by palace intrigue from the start. Just last week, Tom Price resigned as secretary of health and human services after being publicly scolded by Mr. Trump for using chartered flights. Mr. Trump has lost a chief of staff, a national security adviser, a chief strategist, a press secretary and two communications directors. He has fired the F.B.I. director, belittled his attorney general and publicly assailed the deputy attorney general.
That turnover is one reason Mr. Trump has not pushed Mr. Tillerson out, according to advisers. Speaking with reporters in Las Vegas, Mr. Trump dismissed the report that Mr. Tillerson considered resigning. “It was fake news,” he said. “It’s a totally phony story.” Asked about the secretary, Mr. Trump said, “Total confidence in Rex. I have total confidence.”
But even as Mr. Tillerson denied a rift with the president on Wednesday, he alluded to significant differences over North Korea and Iran. He stressed a peaceful resolution of the nuclear dispute with North Korea and associated himself with Mr. Mattis, who just a day earlier endorsed retaining the nuclear agreement with Iran that Mr. Trump has threatened to rip up.
“President Trump’s ‘America First’ agenda has given voice to millions who felt completely abandoned by the political status quo and who felt their interests came second to those of other countries,” he said. “President Trump’s foreign policy goals break the mold of what people traditionally think is achievable on behalf of our country.”
Senator Bob Corker, Republican of Tennessee and the chairman of the Foreign Relations Committee, expressed hope that Mr. Tillerson would not leave because he serves as a check on instability. “I think Secretary Tillerson, Secretary Mattis and Chief of Staff Kelly are those people that help separate our country from chaos, and I support them very much,” he said.
But others said it was time for him to go. “Rex Tillerson has been dealt a bad hand by the Potus & has played it badly,” Richard N. Haass, a State Department official for Republican presidents and now the president of the Council on Foreign Relations, wrote on Twitter, using the initials for president of the United States. “For both reasons he cannot be effective SecState & should resign.”
Mr. Tillerson has been frustrated for months, not just by Mr. Trump’s unpredictable policy positions but by his provocative leadership style. He publicly distanced himself when Mr. Trump blamed “both sides” for violence at a white supremacist rally in Charlottesville, Va., and bristled when the president gave a political speech to the Boy Scouts of America, an organization Mr. Tillerson once headed. NBC reported that he was so offended by the Boy Scouts speech that he threatened not to return to Washington from a visit to Texas.
The episode that Mr. Tillerson called a real breaking point, according to associates, came when he was trying to mediate a dispute between the Persian Gulf state of Qatar and its Arab neighbors. The secretary had long told colleagues that relationships he built over decades in business made him uniquely qualified to broker a deal.
But he complained bitterly that he was undermined by Mr. Trump and Mr. Trump’s son-in-law, Jared Kushner, whom he accused of siding with the United Arab Emirates. When Mr. Tillerson publicly called for a “calm and thoughtful dialogue,” the president less than an hour later lashed out at Qatar as a financier of terrorism.
Mr. Tillerson was also angry with Stephen K. Bannon, then the president’s chief strategist, whom he accused of planting negative news stories, including a report that the secretary had dressed down a White House official, Johnny DeStefano, in the West Wing. During that same meeting, Mr. Tillerson groused to White House officials about Mr. Trump’s tweets, describing them as counterproductive to his efforts at diplomacy, according to a person with direct knowledge.
His aides also have been looking over their shoulders at Nikki R. Haley, the ambassador to the United Nations, who is seen as a probable replacement if Mr. Tillerson leaves. R. C. Hammond, Mr. Tillerson’s spokesman, told NBC that Mr. Pence had asked Mr. Tillerson if Ms. Haley was helpful to the administration.
Mr. Hammond on Wednesday said he “spoke out of line about conversations I wasn’t privy to,” and a spokesman for Mr. Pence said any suggestion that the vice president questioned Ms. Haley’s value was “categorically false.”