Officials told ABC and Fox that Rosenstein had planned to serve as deputy for two years - a milestone that was approaching.
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.
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. This never happened, either, nor did Rosenstein's ouster once Sessions left the Justice Department in November, despite the president tweeting a meme showing several of his political opponents - and Rosenstein - in prison.
On Wednesday, Republican Senator Lindsey Graham said Barr would permit Mueller to complete the investigation into Russia's role in the 2016 presidential election.
However Mr Rosenstein was stripped of the role late a year ago when Mr Sessions was forced out by the president. Rosenstein, who had written a memo supporting the dismissal of former FBI Director James Comey based on the premise that Comey had treated Hillary Clinton unfairly during the 2016 campaign, had more recently clashed with Trump after reports he made comments about overthrowing Trump or secretly recording him shortly before Mueller's appointment.More news: Trump still weighing emergency declaration for border wall: spokeswoman
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Whitaker also was critical or skeptical about the special counsel investigation.
Barr submitted a memo to the Justice Department previous year that argued the law doesn't support an inquiry into whether Trump may have obstructed justice if he tried to frustrate the Russian Federation investigation.
Once confirmed, Barr would assume oversight of the probe.
The department official, speaking on condition of anonymity, said there is no specific time table for Rosenstein's departure and that he plans to leave sometime after Barr's confirmation in the Republican-led Senate.
"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.