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". The other person characterized Barr as "the leading contender".
Barr, however, has loyally supported many of Trump's decisions from the sidelines.
Since leaving the Bush administration, Barr has returned to private practice, including over a decade as the general counsel for Verizon. 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.
The White House declined to comment.
Before that, he also served as deputy attorney general and assistant attorney general overseeing the Office of Legal Counsel.
The former attorney general hasn't shied away from weighing in on Mueller's investigation. He's been mentioned as a potential nominee, and Senator Lindsey Graham, a South Carolina Republican who is in line to lead the Judiciary Committee next year and oversee Barr's confirmation, said he would be a good choice.
Two people familiar with the discussions said the president has told advisers in recent days that he plans to nominate Barr. Patrick Leahy, D-Vt., before Barr's hearing on November 12, 1991.
Barr, he said, could fit that bill.
At the Justice Department, Barr's work ranged from combatting violent crime to investigation the Pan Am 103 bombing.
In a 2017 Fox News interview, Barr said that Mueller could have been "more balanced" in his choice of prosecutors, a majority of which have either ties to Democrats or have made donations to them.
He said there was more basis to investigate a uranium deal approved while Clinton was secretary of state in the Obama administration than potential collusion between Russian Federation and the Trump campaign.
"To the extent it is not pursuing these matters, the department is abdicating its responsibility", Barr said.
"In my view, prosecutors who make political contributions are identifying fairly strongly with a political party", Barr told the Post in July. But some inside the White House were concerned that Barr was too aligned with establishment GOP forces.