Privately, even Mr. Priebus’s critics wondered how Mr. Kelly would surmount the same challenges — controlling a freewheeling president who often circumvents paid staff members by seeking counsel from a roster of outside advisers.

Other aides were left to wonder about their own future. Mr. Trump has considered pushing out Stephen K. Bannon, his chief strategist, according to a White House official who discussed internal deliberations on the condition of anonymity. Several conservative supporters of Mr. Bannon — including Representative Mark Meadows, the House Freedom Caucus chairman — told Mr. Trump on Friday that the president would risk losing base supporters if he let the strategist go.

Mr. Bannon also helped bring Mr. Kelly into the administration during the transition, and was among those who supported his move to chief of staff, an official familiar with his position said.