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.
Officials told ABC and Fox that Rosenstein had planned to serve as deputy for two years - a milestone that was approaching. Lindsey Graham said after meeting with Barr, who led the Justice Department under President George H.W. Bush, that Barr has a "high opinion" of Mueller.
Barr was nominated in December to run the Justice Department after Attorney General Jeff Sessions resigned at the request of Trump, according to a letter delivered to the White House.
If Barr is confirmed, he would likely assume oversight of the Russian Federation investigation, which he has previously expressed criticisms of.
In his time in charge of Mueller's years-long investigation, Rosenstein has been criticized by some Republicans for authorizing the surveillance of former Trump campaign adviser Carter Page, and for failing to appoint a second special counsel to probe his own department's handling of the Russian Federation investigation.
Senate hearings for Mr Barr's confirmation are scheduled to begin on 15 January.
Despite clear tensions between the acting attorney general and the president, an unnamed source familiar with the situation told CNN that Rosenstein is not being forced out of the White House.More news: Apple drops revenue guidance after slowing sales in China
Rosenstein plans to leave at some point after that, though no date has been set and there is no formal plan for the departure, according to the person, who was not authorized to discuss internal conversations publicly and spoke on the condition of anonymity.
She says he wants to help with the transition to a new attorney general.
Barr has told people close to him that he wanted his own No. 2.
But he has been a frequent target of President Donald Trump's criticism.
After Trump fired then-FBI director James Comey in May 2017, legal experts suggested Mueller was pursuing an obstruction of justice case against the president.
Sen. Graham, the incoming chair of the Senate Judiciary Committee, said that Barr would interfere with Mueller's investigation. Chuck Grassley of Iowa, who said he believes Barr's previous experience 'ought to make his nomination very easy'.
Graham said Barr said that if he were attorney general, he would "err on the side of transparency" when he eventually received Mueller's report.