At a news conference in December, Rosenstein said that Mueller's investigation would be "handled appropriately" no matter who is overseeing it.
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.
And Graham effectively confirmed the reporting about the pending Rosenstein departure. Barr would take over control should he be confirmed by the Senate, assuming the same final say over major investigative steps that acting Attorney General Matthew Whitaker has had since November.
Deputy Attorney General Rod Rosenstein is planning to leave the Justice Department shortly after William Barr, the President's nominee for attorney general, is confirmed, according to a source familiar with his thinking.
Sen. Tim Kaine (D-Va.) said Rosenstein's planned departure is going to "really raise the stakes" of Barr's confirmation hearing next week, while Senate Minority Leader Chuck Schumer (D-N.Y.) said Wednesday that he believes Trump should withdraw Barr's nomination.
But he has been a frequent target of President Donald Trump's criticism.
In September, Rosenstein went to the White House expecting to be fired after news reports surfaced that he had discussed secretly recording Trump and invoking a constitutional amendment to remove Trump as unfit for office.
Trump to address Americans amid fears of national emergency
Research also indicates that undocumented immigrants are much less likely to commit crime than native-born American citizens. Some say Trump's hardline policies are slowing processing for migrants, creating an overwhelming bottleneck at the border.
Graham also told reporters that Barr said he wants the report Mueller is expected to produce to be "transparent". Rosenstein insisted he was making a joke.
Trump has denounced the investigation as a witch hunt.
Rosenstein considered resigning in September after reports surfaced he discussed wearing a "wire" to secretly record conversations with Trump in the Oval Office a year ago and discussed the possibility of Cabinet officials invoking the 25 Amendment to remove the president.
In November, the president said Rosenstein "should have never picked a special counsel" to investigate the Russian Federation matter and retweeted an image depicting Rosenstein and several other current and former officials in a prison cell.
Last year, the White House expressed alarm over the warrant to spy on Page, saying it "raises serious concerns about the integrity of decisions made at the highest levels of the Department of Justice and the Federal Bureau of Investigation to use the government's most intrusive surveillance tools against American citizens".
"Deputy Attorney General Rod Rosenstein is out - oh wait, he isn't, but he might be soon".
He has accused the investigation of being led by "Democrat loyalists".