Congressional committees investigating Russia should expect Robert Mueller, the special counsel to probe Russia’s meddling in the 2016 election and other related matters, to tell them to back off, The New York Times reported Thursday night.

Republican senators hinted that Mueller is expected to request that Congress hold off on public hearings with specific witnesses who could be key to the probe.

Congress must “be focused on what our role is,” Texas Republican Sen. John Cornyn, a member of the Intelligence Committee, told The Times. “We’re not the F.B.I. We’re not the Department of Justice. We’re conducting oversight investigations and that’s, I think, our appropriate role.”

South Carolina Republican Sen. Lindsey Graham agreed saying, “You’ve got a special counsel who has prosecutorial powers now, and I think we in Congress have to be very careful not to interfere.”

Despite positive reactions from Democrats regarding Mueller’s appointment, some leaders stressed that the five separate Senate and House committee investigations into the Russia matter should continue.

House Minority Leader Nancy Pelosi said on Wednesday, “A special prosecutor is the first step, but it cannot be the last. Director Mueller will still be in the chain of command under the Trump-appointed leadership of the Justice Department. He cannot take the place of a truly independent, outside commission that is completely free from the Trump Administration’s meddling. A special prosecutor does not negate the need for vigorous Congressional investigations either.”

Democratic Michigan Rep. John Conyers said in a statement, “Let me be clear: the appointment of special counsel does not relieve Congress of its responsibility to conduct oversight of the Trump Administration.”

“I am disappointed that Congress has taken precious few meaningful actions to investigate President Trump’s decision to fire Director Comey, the President’s admission that he did so because he did not like the Russia investigation, and reports that Trump tried to waive Director Comey off of the investigation altogether. I continue to stand with my colleagues on the House Judiciary and Oversight committees in urging our chairmen to look into these matters without delay,” Conyers said.

Follow Kerry on Twitter