If she gets confirmed, it is unclear if Trump will plan to keep the United Nations ambassador as a Cabinet position, a status that would give Nauret more influence in the administration.
The post of attorney general has vexed Mr. Trump since the early days of his presidency, when Mr. Sessions recused himself from overseeing an investigation into possible collusion between Russian Federation and the Trump campaign.
The Washinton Post also quoted former state department official Brett Bruen as saying, "While Haley didn't bring foreign policy experience, she ran a state".
President Trump confirmed the nomination in a tweet.
Trump has railed against the probe and his Justice Department's supervision of it.
Prior to Trump's comments, Barr had emerged this week as a consensus candidate to succeed Jeff Sessions as attorney general, two sources familiar with Trump's thinking told CNN on Thursday.
CBS News' Major Garrett and Fin Gomez contributed reporting.
"This should never again be allowed to happen to a future President of the United States!"
"I look forward to having him join our very successful Administration!" he added on Twitter.
White House aides, speaking on condition of anonymity, said they believe Nauert's key assets include strong communications skills and a fluency with the Trump White House, particularly in understanding the thinking of the president and secretary of state. Bringing a veteran conservative Attorney General who is critical of the Russian Federation investigation but supportive of Trump's anti-Clinton position would also serve Trump well. He is now a lawyer at Kirkland & Ellis.
"To the extent it is not pursuing these matters, the department is abdicating its responsibility", Barr said. That move ultimately led to the appointment of special counsel Robert Mueller, whose long-running investigation has been criticized bitterly by the president.
Watch the clip above, via The Washington Post.
"When one looks at the history of United Nations ambassadors going back to the 1950s, you had people like Adlai Stevenson, who was the Democratic Party nominee for president twice, George HW Bush".
Nauert - a former anchor of "Fox and Friends", among the television-loving Trump's favourite shows - became the spokeswoman of the State Department with no foreign policy experience.
She would replace Nikki Haley, who was twice elected governor of SC but also lacked foreign policy experience. A two-term governor of SC before joining the Trump administration, Haley announced she was stepping down with high approval ratings and a promise to support President Donald Trump in his re-election bid in 2020.
Mr Trump earlier said his daughter, Ivanka, would be flawless for the role but conceded he'd be accused of nepotism if he appointed her.