Nauert, who once was a Fox News reporter and now serves as a spokeswoman for the State Department, was nominated by President Trump to replace outgoing U.N. Ambassador Nikki Haley last Thursday. Trump said, "He was my first choice from day one".
Barr has spent most of his career in the private sector, with two stints at the Justice Department.
The Senate Foreign Relations Committee would consider Nauert's nomination, and one committee member, GOP Sen.
"William Barr's record suggests that he will follow Jeff Sessions' legacy of hostility to civil rights and civil liberties", Faiz Shakir, the ACLU's national political director, said. Barr would now inherit the department which was settled up by Jeff and would implement Trump's tough attitude towards the illegal immigration with some operations against human trafficking, transnational gangs (especially MS-13) and drug trafficking.
While he led major initiatives of the Trump administration to crack down on crime and illegal immigration, Sessions had angered the president early on when he recused himself from overseeing the Mueller probe because of his own contacts with Russians.
In 1989, he was appointed by President Bush to head the Justice Department's Office of Legal Counsel.
Barr said, "I think it is quite understandable that the administration would not want an Federal Bureau of Investigation director who did not recognize established limits on his powers". He now works at Kirkland & Ellis as a lawyer.
Nauert would succeed Haley, a former SC governor who also had little experience in world affairs before she took the job.
Given that report, Trump's latest AG nominee probably won't do Mueller's investigation any good.
"To the extent it is not pursuing these matters, the department is abdicating its responsibility", Barr told the newspaper.
"In my view, prosecutors who make political contributions are identifying fairly strongly with a political party", Barr said.
He has raised concerns about Mr Mueller's appointment of prosecutors with a history of donating to Democratic candidates, saying he "would have liked to see him have more balance on this group".
"I think the president has chosen a superb nominee who is precisely what the Department of Justice needs now, which is a steady hand", said Joseph diGenova, a Trump supporter and former USA attorney.
The Times has described Mr Barr as a lawyer with a vision for broad presidential powers, but also one who sees the attorney general's duty as avoiding political considerations. "She's very talented, very smart, very quick and I think that she will be respected by all", said the President. Secretary Pompeo appointed her as Acting Undersecretary of State for Public Affairs and Public Diplomacy in April this year and she has accompanied Mr. Pompeo on foreign trips, most recently to Brussels on December 3 for a North Atlantic Treaty Organisation meeting of Foreign Ministers and meetings with the Belgian administration as Belgium takes a seat at the UN Security Council.