Trump took to his Twitter account Thursday morning to denounce the report.
Rosenstein, 52, has kept a relatively low profile throughout his decades working inside the Justice Department, a fact that likely helped him win bipartisan approval weeks ago when 94 senators voted to confirm his appointment to deputy attorney general.
There has been no indication that Mr Mueller told the president to fire Mr Comey.
Mark Corallo, a spokesman for Trump's personal lawyer, had responded Wednesday evening to the Post report by saying: "The FBI leak of information regarding the president is outrageous, inexcusable and illegal".
"I recognize the importance of these questions, and I think that Director Mueller ought to review that and make a determination of whether or not he believes it is within the scope of his investigation".
He also told the Senate Intelligence Committee in his June 8 testimony he believed Mr Trump had directed him to drop a related agency investigation into the President's former national security adviser, Michael Flynn.
"Mueller is setting up a dragnet of obstruction, financial questions and every aspect of Trump's life and his associates lives", said former House Speaker Newt Gingrich, who labeled it "very unsafe".
Significantly, Preet Bharara, the India-born former top U.S. federal prosecutor, earlier this week had said that there was enough evidence to begin an obstruction of justice case against Trump over his alleged interference in the Russian Federation probe.
Gunmen Storm Resort In Malian Capital Bamako
One of victims killed in the attack was a French-Gabonese citizen, while the other has not yet been identified, Traore said. Al-Qaida's North African affiliate, al-Qaida in the Islamic Maghreb (AQIM) claimed responsibility for that attack.
Robert Mueller, the special counsel investigating alleged ties between President Donald Trump's campaign and Russian Federation, has hired 13 lawyers to handle the probe, and has plans to add more, according to Peter Carr, his spokesman.
Mueller is investigating Russian interference in USA elections and possible Russian ties to the Trump campaign.
The move ordered Thursday by the transition's general counsel cast a wide net on documents tied to the Russian Federation investigation as well as inquiries into the activities of Trump associates.
"As a result, the Federal Bureau of Investigation is unlikely to regain public and congressional trust until it has a director who understands the gravity of the mistakes and pledges never to repeat them".
Citing a memo from the general counsel's office of Trump's transition team, the Times said members were given the order on Thursday for any information involving Russian Federation or Ukraine in the latest sign of the investigation's expanding reach.
Mueller's probe recently expanded to include obstruction of justice charges against Trump, possible financial crimes of the president's associates and Kushner's business dealings.
Trump also trained his fire on one-time rival Hillary Clinton. But it's part of the President's mindset that he's under siege and he's going to fight like he always has.
Mueller, who was appointed as special counsel by Deputy Attorney General Rod Rosenstein on May 17, is investigating possible coordination between the Kremlin and the Trump campaign to influence the 2016 presidential election, and related matters. "For the love of god, can we get back to the original thing that was proposed, the independent commission, and get away from all these entanglements?" he said.