Other options range from the possible — including Mick Mulvaney, the director of the Office of Management and Budget — to the unlikely, including Chris Christie, the former New Jersey governor and head of the Trump transition team, who angered Mr. Kushner for sending his father to jail while he was a federal prosecutor.
Mr. Trump, who enjoys embarrassing reporters and planting his own information about staff machinations, has mentioned several other names. They included Matthew G. Whitaker, the current acting attorney general who nearly every West Wing staff member has said could not be a realistic option, but who Mr. Trump likes personally.
Some White House aides made the same point about Mr. Ayers, who has never faced extensive scrutiny of his finances or his previous political work. They said that those questions were almost certain to come if he ascended to the higher job.
That included not just his personal fortune — worth somewhere from $12.2 million to $54.8 million — but also his tenure as executive director of the Republican Governors Association and working for Eric Greitens, the former governor of Missouri, who resigned in disgrace after a woman accused him of physically abusing her during a nonconsensual relationship.
Mr. Kushner wants Mr. Ayers to take over running the “super PAC” backing the president, which people close to Mr. Trump believes needs the kinds of fund-raising boost that Mr. Ayers can provide. Such a move would keep him in the Trump orbit, which Mr. Trump’s children in the White House want.
Mr. Kelly, meanwhile, is said to be furious with Ms. Trump and Mr. Kushner. One senior administration official said that Mr. Kelly was known to have kept written notes about Mr. Kushner and Ms. Trump and the things that they had done or requested, which he conspicuously left on his desk in view of his staff.
Mr. Trump, who was said to be distrustful of Mr. Kelly’s personnel judgment, will not be consulting Mr. Kelly on his successor, people close to him said. Still, Mr. Kelly returned to the White House in his role on Monday, and attended a meeting with the president.