In this November 26, 1991, file photo, President George H.W Bush, right, and William Barr wave after Barr was sworn in as the new Attorney General of the United States at a Justice Department ceremony in Washington.
Barr has criticised Mueller for hiring attorneys for his team who contributed to Democratic politicians, including Hillary Clinton.
Even if Barr were announced as the president's choice this week, it could take months for a confirmation vote, given the congressional schedule.More news: Christchurch backs Theresa May's Brexit deal
In the early 1990s, as attorney general, Barr supervised Robert Mueller in his work as the chief prosecutor for the Department of Justice criminal division.
Trump has been known to change his mind on key personnel decisions before announcing them.
Be sure to read Allahpundit's analysis on Barr from yesterday, when the Washington Post profiled him as the man most likely to succeed Sessions. He is now a lawyer at Kirkland & Ellis and does work advising corporations on government enforcement and regulatory actions.
The Mueller probe has been an Achilles heel for the office of attorney general the past couple of years.
This, of course, is no normal time - and it didn't take long for interested parties to scour Mr Barr's recent comments and writings in search of any meaningful hints on how he would oversee Robert Mueller's Russian Federation investigation.More news: New Game Of Thrones Season 8 Teaser Points To An Epic Showdown
To that end, in reference to the Uranium One deal that gave a Russian state-backed firm control of 20% of US uranium production, Barr said there is more of a reason to look into that deal than possible collusion between Russia and the Trump campaign.
"To the extent it is not pursuing these matters, the department is abdicating its responsibility", Barr said.
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.
Barr added that he "would have liked to see him have more balance on this group".
Barr also said James Comey's firing was "understandable" and that he "crossed a basic line" and "transgressed" in his role as Federal Bureau of Investigation director.More news: Microsoft confirms Edge is moving to Chromium
Senator John Cornyn of Texas, the No 2 Republican leader, agreed.