Barr's Senate confirmation hearing is scheduled to begin January 15, which means a vote would be held mid-February at the earliest, CNN reported.
Sources told ABC News Rosenstein wants to ensure a smooth transition to his successor and would accommodate the needs of Barr, should he be confirmed. Rosenstein and his chief deputy have continued to maintain day-to-day oversight over the probe, a senior Justice Department official told reporters last month.
Mueller is investigating Russia's meddling in the 2016 election and contacts with the Trump campaign.
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. "My guess is that he is making room for the new attorney general to build a team that he wants around him", Sanders said.
After the report broke, Trump promised to get rid of a "lingering stench" at the Justice Department, but weeks later stated that he had a "very good relationship" with Rosenstein, and had "no plans" to give the Deputy AG his marching orders.
ABC News also reported that there is no indication that Trump is attempting to force out Rosenstein, despite their rocky past.
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.