Barr's confirmation as attorney general ― a position now occupied on an acting basis by Matthew Whitaker following the forced resignation of Jeff Sessions ― would ensure a smooth transition, the person said.
A source close to Rosenstein told NBC that the deputy attorney general's departure hinges specifically around the report Mueller will soon file on his sprawling investigation into Russia's interference in the election.
Rosenstein was said to have discussed removing Trump from office or secretly recording conversations with him; he said he never authorized any action after those conversations but Justice Department sources said he expected to be fired.
If Barr's chances for confirmation appear strong, Democrats nonetheless are expected to question him sharply about the skepticism he has voiced about the Russian Federation investigation.
Rosenstein has likely been serving on borrowed time since a New York Times report that described conversations he had with people inside the Justice Department early in his tenure.
The source said Rosenstein is not being forced out, and he has conveyed his thinking to the White House.
Trump has frequently railed against the deputy attorney general on Twitter, with the president recently retweeting an image depicting Rosenstein and others behind bars.
Rosenstein's willingness to leave DOJ could be read as another sign that the Mueller probe is winding down ahead of an anticipated report laying out the evidence investigators have uncovered about Russian election interference and the Trump campaign.
Spokeswoman Sarah Huckabee Sanders said Wednesday Rosenstein had always planned to stay around two years and wants to help with the transition to a new attorney general.
CNN said Barr told the White House he wanted his own deputy.
"I asked Mr. Barr directly do you think Bob, Mr. Mueller's on a witch hunt?"
However Mr Rosenstein was stripped of the role late a year ago when Mr Sessions was forced out by the president.
Senate hearings for Mr Barr's confirmation are scheduled to begin on 15 January.
He then handed off oversight to his deputy, putting Mr Rosenstein in the spotlight and leading to criticism from Republicans in Congress as well as Mr Trump.
Rosenstein remained in the post after Trump named political loyalist Matt Whitaker as acting attorney general in November. But Democrats will use the hearing as an opportunity to extract commitments from the nominee about his approach to the Mueller investigation.