According to the Washington Post, former attorney general William P. Barr is the leading candidate to head the Justice Department. Barr did not respond to ABC's request for comment.
If confirmed by the Senate, Barr would succeed Attorney General Jeff Sessions, who was forced out by Trump in November.
Trump says that Barr is "a terrific man" and "one of the most respected jurists in the country" and says, "I think he will serve with great distinction".
Barr has said there is more reason to investigate potential wrongdoing by Trump's campaign opponent, Democrat Hillary Clinton, than there is to probe any potential collusion.
"While the FBI carries out investigative work, the responsibility for supervising, directing and ultimately determining the resolution of investigations is exclusively the province of the Justice Department's prosecutors".
Trump's consideration of a new attorney general comes during a critical time for special counsel Robert Mueller's investigation, of which the next Justice Department chief will inherit oversight.
In a 2017 opinion piece for the Post, Mr Barr said Mr Trump had been right to fire FBI director James Comey because he "crossed a line" by announcing the outcome of the investigation into Mrs Clinton's emails.
Sen. Dianne Feinstein, the top Democrat on the Judiciary Committee, said she was unfamiliar with Barr's views but encouraged by a suggestion that he was an "establishment" Republican.
Barr added that he "would have liked to see him have more balance on this group".
In an administration rife with internal conflict and deeply distrustful of the UN, Nauert's nomination would place a less senior person at the worldwide agency than Haley, who reportedly sparred with other administration officials.
One of the people who spoke to the AP said there have been discussions among senior administration officials about Barr's willingness to do the job, and said the belief was that he was open to doing it if asked.
Senator John Cornyn of Texas, the No 2 Republican leader, agreed.