But with Mr. Trump, a list of finalists is sometimes only a starting point, and he has been known to consider other candidates as names are floated to him or he happens to spot potential advisers on television. Some on the list he provided on Tuesday may be on there only to stroke egos or throw off those trying to divine his thinking.
Soon after Mr. Trump’s comments to reporters, the White House press secretary, Stephanie Grisham, clarified that the five names were “not the full list, and there are others being considered.”
None of the people Mr. Trump named are considered strong ideologues — with the exception of Mr. Fleitz, a hawk in the mold of Mr. Bolton, his former boss. But Mr. Trump has passed over Mr. Fleitz before, declining to install him as director of national intelligence to replace the departed Dan Coats, and is not expected to tap him this time.
Ms. Gordon-Hagerty, a highly regarded veteran of nuclear security policy, was suggested to Mr. Trump by his daughter Ivanka after he asked for female candidates, but is seen at the White House as another long shot.
Whomever Mr. Trump eventually picks would be his fourth national security adviser in less than three years in office, more than any other president has had in a first term. He fired his first, Michael T. Flynn, a retired Army lieutenant general, after only 24 days for misleading Mr. Pence and others about his interactions with Russia’s ambassador. He pushed out his second, Lt. Gen. H. R. McMaster, after the relationship soured.
Mr. Bolton left last week amid deep disagreements on how to handle issues like Iran, North Korea, Russia and Afghanistan. The two could not even agree on the circumstances of his departure. Mr. Trump announced that he had fired him, but Mr. Bolton insisted he offered his resignation without being asked.
For now, the National Security Council is being run by Mr. Bolton’s onetime deputy, Charles M. Kupperman, a former defense industry executive who served in the Reagan administration. The council’s senior director for the Middle East, Victoria Coates, remains in her job.