Yet friends and even family members have had difficulty getting through the tight protective screen that Mr. Kelly has built around the president. From some of those family members, Mr. Trump recently learned that his chief of staff has canceled meetings with some of their old friends, without letting Mr. Trump know first, according to three people familiar with what happened.

Ivanka Trump, the president’s daughter, has at various times expressed frustration with Mr. Kelly, according to people familiar with the conversations. Other White House aides have said that Mr. Kelly has at times appeared dismissive of her in meetings or on email exchanges with other advisers, and has sought to limit the portfolios of both her and her husband, Jared Kushner. Mr. Kelly has struggled to handle the couple, who are in the unusual position of being official members of the staff and who have hosted policy dinners at their homes. But Mr. Kelly has insisted that he has no personal animus toward them, according to one White House aide.

Ms. Trump has also complained to colleagues that people try to use her as a cudgel against Mr. Kelly, imparting a level of anger at him that she says she herself does not feel.

In a statement on Thursday, she denied that she had issues with Mr. Kelly and praised the job he was doing as chief of staff.

“I typically ignore false rumors spread by bad actors in an attempt to sow internal division,” she said, “but I feel compelled to respond and say that it is an honor to work for Chief Kelly, who is doing a fantastic job. There exists a deeper bond among the team today than at any point prior.”

Mr. Kelly did not travel with the president to Davos, a fact that elicited suspicion about whether he was being excluded in the same way that Mr. Priebus was before he was eventually pushed out. But a senior White House official said that Mr. Kelly had decided before Wednesday’s impromptu gaggle to remain behind because he wanted the time to consider his strategy for the next few weeks. On Tuesday night, Mr. Trump gives his State of the Union address, a chance to set out his agenda before a huge prime-time audience.

Names of prospective replacements for Mr. Kelly, including members of Congress and the cabinet, have been bandied about outside the White House. Mr. Trump has not engaged in the speculation — at least, not yet.