Rosenstein appointed Mueller as special counsel in May 2017 to investigate potential coordination between Russian Federation and the Trump campaign to sway the 2016 election.
In October, speculation mounted that Trump would fire Rosenstein, after a salacious New York Times op-ed by a supposed Trump cabinet "insider" claimed that Rosenstein had suggested wearing a wire to secretly record the president to prove he was unfit for office.
Trump's critics have expressed concern that Barr may try to curtail Mueller's investigation, which Trump repeatedly has called a "hoax" and a "witch hunt".
In June, Barr sent an unsolicited memo to the Justice Department questioning Mueller's decision to look into whether the president obstructed justice when he asked then-FBI Director Comey to stop a probe into former national security adviser Michael Flynn over his communications with Russian officials, according to The Wall Street Journal.
United States jobs growth exceeds expectations and eases market fears
Market Pulse Stories are Rapid-fire, short news bursts on stocks and markets as they move. Former treasury secretary Henry Paulson said that China "is under a microscope".
Spokeswoman Sarah Huckabee Sanders says Rosenstein had always planned to stay around two years. "My guess is that he is making room for the new attorney general to build a team that he wants around him".
"I can assure you he has a very high opinion of Mr. Mueller, and he is committed to letting Mr. Mueller finish his job", Graham told reporters after the meeting.
He also wanted to help ensure Barr, if confirmed by the Senate, had a smooth transition, ABC said, citing sources.
But he has been a frequent target of President Donald Trump's criticism. Barr, a conservative steeped in the law-and-order politics of the early '90s, began meeting with senators on Capitol Hill on Wednesday ahead of his confirmation hearing scheduled for Tuesday. Barr and Mueller worked together when Barr was Bush's attorney general between 1991 and 1993 and Mueller oversaw the department's criminal division.
Republicans, of course, control the Senate with 53 votes, so ― barring some huge revelation ― Barr's confirmation doesn't appear to be in any real jeopardy. Barr would most likely take over that role, since Rosenstein only assumed it after former Attorney General Jeff Sessions recused himself. But Democrats will use the hearing as an opportunity to extract commitments from the nominee about his approach to the Mueller investigation.