Mr. Kelly himself is also on thin ice, according to officials in the White House. He is said to have angered the president by privately saying “no” to the boss too often.

Mr. Trump grew frustrated with Mr. Kelly, in part for stalling on the president’s desire for tariffs. The president finally forced the issue a few weeks ago, announcing tariffs on steel and aluminum, which prompted Gary D. Cohn, his top economic adviser, who had opposed the policy, to announce his resignation.

In recent days, Mr. Kelly has sought to repair his standing with the president by moving more aggressively to address issues that have festered, including staffing concerns at the White House and cabinet secretaries with political liabilities.

Several high-level people in the administration are at risk.

Mr. Carson and David J. Shulkin, the secretary of veterans affairs, have both embarrassed the president by generating scandalous headlines. Mr. Carson could be cut over an eye-popping $31,000 dining set, and Mr. Shulkin might be replaced over a 10-day, $122,000 European trip with his wife.

And then there is Attorney General Jeff Sessions, whose original sin — the decision to recuse himself from oversight of the Russia investigation — made him the regular target of presidential ire. The attorney general has threatened to resign at least once, but has more recently indicated his determination to resist Mr. Trump’s obvious desire for him to leave his post at the Justice Department.

Mr. Trump could act as early as Friday to remove one or more of them, though the president is known to enjoy keeping people off kilter.

In his remarks on Thursday, Mr. Trump assailed predictions of further staff shake-ups by saying such reports were “a very exaggerated and false story.” But he hinted that his choices for a cabinet might have been different had he known then what he knows now.