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.
William Barr served as attorney general during the late President George H.W. Bush's administration.
If Mr Barr is confirmed by the senate after his official nomination, he will replace the current acting attorney general, Matt Whitaker.
Sources familiar with the matter told the Post that Barr is a "really serious contender, and possibly the front-runner" for the position of attorney general.
Barr was the general counsel of GTE when I represented that company (as one of dozens of lawyers from at least four law firms to do so) in contentious litigation following the enactment of the Telecommunications Act of 1996. "(Barr) may be the kind of person who would be easier to confirm".
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.
Mr Barr retired from his position as counsel and executive vice-president of Verizon in 2008.
Barr served as Mueller's boss when he was attorney general in the early 1990s and Mueller led the criminal division at the Justice Department.
Barr also wrote past year that the administration's decision to fire James B. Comey as FBI director was "quite understandable" because, in his view, Comey had usurped the power of the attorney general when he publicly announced his recommendation not to charge former Secretary of State Hillary Clinton during the investigation of her private email server.
Prior to joining the Justice Department, Barr worked with the White House Domestic Policy staff for a year under then-President Ronald Reagan.
Richard Cullen, who served as U.S. Attorney under Barr, describes him as a good pick for the role.
Still, while in private practice, Barr has occasionally weighed in on hot-button investigative matters in ways that could prompt concerns among Democrats. In 2017, he told The Washington Post that "prosecutors who make political contributions are identifying fairly strongly with a political party".
Barr, a Columbia University graduate, worked at the Central Intelligence Agency while attending the George Washington University Law School at night. He already received a bachelor's degree in government and a master's degree in government and Chinese studies - both from Columbia University.
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.