Among the many people agitated this week over John F. Kelly, the White House chief of staff, was President Trump. And among the people the president called to express dissatisfaction, according to those close to him, was none other than Reince Priebus, the previous chief of staff, who also irritated Mr. Trump…

Mr. Trump has recently asked advisers what they think of Mick Mulvaney, who currently holds twin posts as director of the White House budget office and the Consumer Financial Protection Bureau, as a possible chief of staff, according to two people briefed on the discussions…

Two people close to the situation said Mr. Kelly had strongly urged Mr. Porter to remain in his job. But a third person denied that, saying that Mr. Kelly had suggested to Mr. Porter on Tuesday night that he should resign. All three agreed that Mr. Kelly never demanded the resignation.