If confirmed by the Senate, Barr will take over for acting attorney general Matthew Whitaker.
One source explained that Kelly, who is the biggest White House player to meet with the special counsel, had "ground rules" established by White House Counsel Emmet Flood for the interview.
The vice chairman of the Joint Chiefs is Air Force Gen. Paul Selva, whose term expires in July 2019.
Compounding the rumors were reports surrounding Kelly taking the day off on Friday, as the president traveled to Missouri. The lights were off in his West Wing office. He was photographed looking grim-faced in the lobby of Trump Tower as the President declared there were "good people" on both sides of the racist violence.
Kelly is the latest high-ranking White House official known to have given information for Mueller's probe.
One senior administration official said Friday that it's clear Kelly will be leaving, though it's not certain that the departure was imminent as CNNreported.
Like all of those interviewed, the person spoke on condition of anonymity to discuss sensitive personnel matters.
President Trump is expected to nominate General Mark Milley, who now serves as Army Chief of Staff.
Kelly reportedly intervened to stop Trump firing Department of Homeland Security Secretary Kirstjen Nielsen last month. Ayers, a sharp-elbowed and ambitious Georgia operative, fits the bill, advisers said.
Mr. Trump re-nominated him in 2017 for a second term. But President Donald Trump is apparently treating the retired Marine general like he's already gone. Trump has chafed at Kelly's management style and resisted some of his moves to instill discipline in the West Wing and contain chaos.
Kelly has seen his status as chief of staff diminished in recent months, with the President circumventing numerous policies and protocols the retired Marine Corps general put in place when he entered the West Wing past year.
Trump has been advised to bring on a chief of staff better suited to and familiar with the rough and tumble of a re-election campaign.
Trump often vacillated between criticizing and praising Kelly, sometimes within minutes of each other. But he has a number of detractors who have been trying to convince the president he'd be a disaster.
Nauert, 48, joined the State Department previous year with no government experience after a career as an anchor and correspondent at Fox News.
Trump hinted the announcement would be coming earlier Friday when he said he would be making an announcement at the Army-Navy football game he will be attending Saturday in Philadelphia. Aides on the Senate Foreign Relations and Judiciary committees - which would take up the ambassador and attorney general nominations, respectively - said Friday that neither committee will hold confirmation hearings this year, given the limited time left before a new Senate takes over on January 3.