From the start, Mr. Bannon, 63, has told people in his orbit that he never expected to last in his current position longer than eight months to a year, and hoped to ram through as much of his agenda as he could while he stood in the president’s favor. More recently he has told friends that he is working in the White House one day at a time, and constantly asks himself whether he could better pursue his to-do list — including cracking down on legal and illegal immigration — on the outside.

But the choice might not be his. At a recent dinner at the White House with Mr. Kushner and Mr. Kelly, before Mr. Trump decamped for a working vacation at his private golf club in Bedminster, N.J., the president listened while one of the guests, Mr. Murdoch, a founder of Fox News, said Mr. Bannon had to go.

Mr. Trump offered little pushback, according to a person familiar with the conversation, and vented his frustrations about Mr. Bannon. Mr. Murdoch is close to Mr. Kushner, who has been in open warfare with Mr. Bannon since the spring.