If there was any doubt that Mr. Kelly faces a difficult balancing act, the chief’s first task on Thursday was to beat back rumors that his own job is in immediate jeopardy. In doing so, he became the second top Trump official in a week to insist he is not about to be fired.

“I would just offer to you that although I read it all the time pretty consistently, I’m not quitting today,” Mr. Kelly quipped, referring to news reports that have suggested he is discouraged at the White House and on the outs with Mr. Trump. “I just talked to the president — I don’t think I’m being fired today. And I’m not so frustrated in this job that I’m thinking of leaving.”

Last week, Secretary of State Rex W. Tillerson faced a similar task after reports that he had called Mr. Trump a “moron.” But where Mr. Tillerson was awkward and brusque during a similar face-off with the press, Mr. Kelly was smooth and genial.

His only frustration, Mr. Kelly said, was repeatedly reading untrue stories about things Mr. Trump said, or of people who are about to be fired.

“That’s my frustration,” he said, “and I mean no disrespect to you all.”

Mr. Kelly also sought to avoid adding to the week’s clash between Mr. Trump and Senator Bob Corker, Republican of Tennessee. Asked about Mr. Corker’s harsh comments about the president, Mr. Kelly carefully gave a nod to Mr. Trump’s anger.

“There’s others that are, as the president will say, grandstanding,” the chief of staff said. “I’m not saying Senator Corker is that way; I’m just saying that some people grandstand and kind of enjoy the attention.”