Barr was George H.W. Bush's attorney general between 1991 and 1993.
The Mueller probe has been an Achilles heel for the office of attorney general the past couple of years.
That person said those advising the president viewed Barr as someone who knows the department well, and is a good manager.
If confirmed by the Senate, he would replace acting Attorney General Matthew G. Whitaker, who took over in November for fired Attorney General Jeff Sessions.
If nominated, media reports said Republicans would likely respect Barr's previous experience and Democrats would view Barr as a Republican lawyer with no particular loyalty to Trump.
As far as what we could expect from an Attorney General Barr in 2019, one can look to a piece he co-penned for the Washington Post with other former AGs in support of former Attorney General Jeff Sessions.
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He would take over management of Special Counsel Robert Mueller's investigation into Russian interference in the 2016 election and possible collusion by Trump's campaign - the issue that cost Sessions his job. Mr. Barr, 68, is a former general counsel and executive vice president of Verizon Communications who works now as a lawyer at the firm of Kirkland & Ellis.
In an administration rife with internal conflict and deeply distrustful of the UN, Nauert's nomination would place a less senior person at the global agency than Haley, who reportedly sparred with other administration officials.
He also defended Trump's firing of Federal Bureau of Investigation director James Comey in an op-ed, and has said special counsel Robert Mueller's team lacks balance.
He was quoted two months later in a Post story as expressing concern that members of Mueller's team had given contributions to Democratic candidates.
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. "I would have liked to see him have more balance on this group", he said in July 2017.
Senator John Cornyn of Texas, the No 2 Republican leader, agreed.