The contrast was on sharp display when Ms. Haley chaperoned the 14 members of the Security Council on a visit to the White House on Monday. Neither Mr. Tillerson nor any of his aides were present, with a spokesman explaining that his schedule “did not enable him to participate.”

Mr. Tillerson has skipped meetings with world leaders at the White House too, though he continues to spend considerable time with President Trump. His defenders say that Mr. Tillerson is confident of the president’s support and that Mr. Tillerson does not believe he needs to appear in front of reporters or TV cameras to confirm his place as the nation’s chief diplomat.

“Any notion that there’s some kind of competition between Haley and Tillerson is laughable,” said James J. Carafano, a Heritage Foundation fellow and a member of the Trump transition team. “She’s filling a role and is comfortable in that role, and I don’t think Tillerson feels threatened by that.”

Ms. Haley has by no means replaced Mr. Tillerson as the administration’s preferred voice on foreign affairs, according to a top White House aide who spoke on the condition of anonymity because of the sensitivity of the subject. Indeed, some in the White House see her as seeking a bit too much attention as the administration formulates its foreign policy, the official said, noting that it is not seen as a problem that needs an immediate solution. The White House is riven by feuds, with various cabinet members vying for prominence, and while Mr. Trump has a business-centric reverence for Mr. Tillerson, he is said to like Ms. Haley.

There was a hint of that tension on Monday during the White House lunch for the Security Council diplomats.

“Now, does everybody like Nikki?” the president asked his lunch guests. “Because if you don’t, otherwise, she can easily be replaced.”