The president and senior White House officials were debating when and how to dismiss Mr. Bannon. The two administration officials cautioned that Mr. Trump is known to be averse to confrontation within his inner circle, and could decide to keep on Mr. Bannon for some time.

As of Friday morning, the two men were still discussing Mr. Bannon’s future, the officials said. A person close to Mr. Bannon insisted the parting of ways was his idea, and that he had submitted his resignation to the president on Aug. 7, to be announced at the start of this week, but it was delayed in the wake of the racial unrest in Charlottesville, Va.