WASHINGTON — Long-simmering tension within President Trump’s national security team spilled into public view on Wednesday as Secretary of State Rex W. Tillerson took the extraordinary step of calling a news conference to affirm his support for Mr. Trump, despite what associates describe as his deep frustration with the president and talk of resignation.

Mr. Tillerson praised Mr. Trump but did not deny a report that he once referred to the president as a “moron.” Mr. Trump welcomed Mr. Tillerson’s statement of support and declared “total confidence” in his secretary of state.

If Mr. Tillerson had hoped to douse questions about how long he would stay, he instead further fueled a debate about his future. Although he insisted he had never considered resigning, several people close to Mr. Tillerson said he has had to be talked out of drafting a letter of resignation on more than one occasion by his closest allies, Defense Secretary Jim Mattis and John F. Kelly, the White House chief of staff. And they said he has regularly expressed astonishment at how little Mr. Trump understands the basics of foreign policy.

A former chief executive of Exxon Mobil, Mr. Tillerson has never found his place as a subordinate to the hard-charging, unpredictable president. He has bristled at being undercut, as he was over the weekend when Mr. Trump publicly said Mr. Tillerson was “wasting his time” by trying to open talks with North Korea. At the same time, Mr. Tillerson has alienated lawmakers, foreign policy veterans and the news media while demoralizing the State Department, and critics inside and outside the White House consider his troubles self-inflicted.