In September the New York Times reported, citing anonymous sources, that Mr Rosenstein had discussed recruiting cabinet members to invoke the 25th amendment of the United States constitution, which provides for the removal of a president if he is deemed unfit for office.
Rosenstein will stay on to ensure a smooth transition with Barr, an official told Reuters news agency, adding that he saw his job as deputy as a two-year stint and is not being forced out.
Barr's Senate confirmation hearing begins January 15, which means a confirmation vote at the earliest would occur in mid-February.
Barr's nomination is likely to meet heavy scrutiny regarding the ongoing investigation, particularly from Democrats, following reports he had written a memo in June questioning the probe.
Barr submitted a memo to the Justice Department past 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.
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Rosenstein was the person who oversaw the probe by special counsel Robert Mueller for more than a year.
If confirmed, Barr would replace appointed acting Attorney General Matthew Whitaker and would assume oversight of the Mueller investigation.
ABC News reported that Rosenstein has told Trump and others that he plans to depart in the coming weeks, probably once the Senate confirms a new Justice Department chief.
It is unclear how long after Barr's appointment Rosenstein is expected to remain within the DOJ.
At a news conference in December, Rosenstein said that Mueller's investigation would be "handled appropriately" no matter who is overseeing it.
The official, who asked not to be named since no announcement has been made, said there is no specific plan for Rosenstein's departure and that he plans to leave sometime after Barr's confirmation.
Rosenstein himself has sought to allay fears about the effect of any one leader on the workings of the Justice Department generally and the Russian Federation investigation specifically.