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(CNN) —  

Deputy Attorney General Rod Rosenstein doesn’t see a need at this point for a special prosecutor in the probe into Russian meddling in the 2016 election, people familiar with his thinking tell CNN.

Democratic lawmakers and others have pushed for the move in the wake of the controversy over President Donald Trump’s firing of former FBI Director James Comey. Critics of the Trump administration’s handling of the Russia investigation have long held the view that a special prosecutor is needed.

The investigation is led by Dana Boente, the US attorney in Alexandria, Va., who also serves as the head of national security prosecutions at Justice Department headquarters.

One source says Rosenstein isn’t inclined to make a change unless the FBI investigation appears to be imperiled. And at this point, FBI officials are confident that the investigation is moving ahead, despite Comey’s abrupt firing earlier this week.

Indeed, Rosenstein has privately vowed to lawmakers and staff that he plans to allow the bureau’s investigation to move forward, uninhibited by pressures from the White House.

At a closed-door meeting Thursday with the two leaders of the Senate intelligence committee, Rosenstein and Boente laid out how the FBI plans to work with the panel, trying to ensure its investigation doesn’t conflict with that of the committee as administration officials and lawmakers interview witnesses and obtain documents on their parallel investigations.

The deputy attorney general also believes that there’s nothing he has seen at this point that would require him to recuse himself from the role overseeing the probe led by Boente, the people familiar with his thinking say. Attorney General Jeff Sessions, who was involved in firing Comey, recused himself from any role in the investigation in early March.

A Justice Department spokeswoman declined comment.

Rosenstein, on the job for just more than two weeks after a 94-6 Senate confirmation vote, has drawn criticism because the White House initially pinned the firing of Comey on Rosenstein, saying he had penned a memo expressing concern about the way Comey had handled the Hillary Clinton email investigation.

The White House later changed its story of how the firing came about, with the President saying in an NBC interview that he planned to fire Comey no matter what Rosenstein and Sessions said.

Rosenstein has told officials he was unhappy with the manner in which the Comey firing took place.

The former FBI director was addressing employees in Los Angeles when a television screen tuned to Fox News nearby flashed a headline incorrectly saying Comey had resigned. He thought it was a joke. He then saw CNN reporting that he was fired and realized it was true.

Sen. Mark Warner, the ranking Democrat on the intelligence panel, said Friday that he expressed at his meeting with Rosenstein and Boente his preference that Rosenstein name a special prosecutor. But Rosenstein made no commitments to him and the committee’s chairman, Richard Burr, Warner said.

“This is where, again, the chairman and I just disagree in terms of the needs for this narrowly tailored independent counsel,” Warner said. “I expressed that concern to Mr. Rosenstein. He took it under advisement.”

Asked to expand on Rosenstein’s thinking, the Virginia Democrat added: “I think he listened.”

In an interview with CNN on Friday, Warner would not say if he has confidence in Rosenstein to oversee the Russia investigation – and he had a blunt warning to Rosenstein if he does not appoint a special prosecutor.

“If he doesn’t do that, then I think it’s going to be very difficult to solicit a lot of support from Democrats … in terms of whoever the President picks to be a permanent FBI director,” Warner said.