The president, who is in the middle of a re-election fight, has grown more resigned and afraid about what the future might hold for his son, according to people close to them.
In a conversation with a woman posing as a conservative supporter, Martha-Ann Alito appeared to push back against having to look at a symbol of L.G.B.T.Q. rights.
Judge Aileen Cannon threw out one basis for the case against the former president, involving a highly sensitive military map he showed an aide after leaving office.
The steps under consideration include protecting them from deportation and providing access to work permits, according to three officials with knowledge of the discussions.
Mr. Burgum, North Dakota’s governor, is a prime contender in the former president’s search for a running mate, but he remains untested on the national stage.
By President Biden’s account, he was a fierce civil rights activist in his youth, had an uncle who fell prey to cannibals and used to drive an 18-wheeler.
Republicans and Democrats alike in Cochise County, Ariz., a place where many have long felt their needs are ignored, eyed the executive order with skepticism.