Ahead of Barr's Senate confirmation hearing next week, top Justice Department officials have sought to downplay his previous argument that the president can't be investigated for obstructing justice.
According to Fox News and ABC News, Mr Rosenstein had long planned to leave after two years as deputy attorney general.
A former USA attorney, Rosenstein expected to be fired back in September 2018 after a report surfaced that he offered to wear a wire during conversations with Trump. Lindsey Graham he would allow special counsel Robert Mueller's investigation to continue unabated, Graham said Wednesday.
Barr told the SC senator that he would follow Department of Justice (DOJ) procedure while "erring on the side of transparency" about sharing Mueller's final report with Congress and the public.
Barr, Graham said, told him he didn't think Mueller was on a "witch hunt", as Trump has branded the Mueller probe.
Barr, who served in the position in the early 1990s and is US President Donald Trump's pick to do the job again, has a confirmation hearing before the Senate Judiciary Committee next week and could be in place at the Justice Department as soon as February.
On Wednesday, White House press secretary Sarah Huckabee Sanders issued remarks about the reports of Rosenstein's purported departure.
She says he wants to help with the transition to a new attorney general. Barr and Mueller worked together when Barr was Bush's attorney general between 1991 and 1993 and Mueller oversaw the department's criminal division.
Rosenstein has had oversight of the US Special Counsel's probe into alleged Russian interference in the 2016 presidential election and possible connections to Trump's campaign.
In this June 21, 2017, file photo, former FBI Director Robert Mueller, the special counsel probing Russian interference in the 2016 election, departs Capitol Hill following a closed door meeting in Washington.
After an embattled turn in that role, Sessions stepped down at the end of a year ago and was replaced temporarily by Matthew Whitaker as Acting Attorney General. Chuck Grassley of Iowa, who said he believes Barr's previous experience 'ought to make his nomination very easy'.
If confirmed, Barr would be in charge of Mueller, unless he too recused himself.
"I'm going to do a deep dive into the FISA issue, I think he'll be part of it", Graham said of Barr.