In a separate news report on Friday, the Washington Post claimed that a senior White House adviser was a person of significant interest in the investigation into possible ties between Trump's election campaign and Russian Federation.
Former FBI director James Comey has agreed to publicly testify before the Senate Intelligence Committee about his role in the investigation of Russian interference in the 2016 election, the committee announced Friday.
The report does not say who the cover-up probe will target.
Several news reports have emerged in the wake of Comey's firing, alleging that Trump demanded "loyalty" from the former FBI director, asked him to "let" the investigation into former national security adviser Michael Flynn go, and told Russian officials that Comey was "crazy" and a "nut job".
"Jarrett writes at Fox News: "[Mueller] and James Comey are good friends and former colleagues who worked hand-in-hand for years at the Federal Bureau of Investigation. "He was insane, a real nut job", Mr Trump said, according to the document, which was read to The New York Times by a United States official.
Even if the Justice Department granted a waiver, the White House would consider using the ethics rule to create doubt about Mueller's ability to do his job fairly, the sources said.
Contacted by the Times, White House press secretary Sean Spice did not deny Trump had made the statements, saying Comey's "grandstanding and politicizing" of the Russian Federation probe had put "unnecessary pressure on our ability to engage and negotiate with Russian Federation".
Robert Mueller, who served as FBI chief under George W Bush, was appointed this week as special counsel for the investigation.
"I faced great pressure because of Russian Federation". "Congress is going to want to look over the shoulder of this investigation. the executive branch will always try to limit that for fear that it would contaminate a potential criminal investigation for leaks, all the while sometimes leaks occur in the executive branch".
Panel Chairman Richard Burr and Vice Chairman Mark Warner said they will schedule the hearing for after Memorial Day.
In the remarks, Rosenstein said he learned a day before Comey's firing that Trump wanted to dismiss him.
The day after he was sacked, officials say Comey had just requested more resources to pursue his investigation into Russia's interference in the election, and whether there was any collusion between Donald Trump's campaign and the foreign power, which fueled concerns that Trump may have been trying to undermine an investigation that could threaten his presidency.
The president's son-in-law Jared Kushner is among those whose contacts with the Russian government have come under scrutiny.
A Washington Post investigation revealed someone close to President Trump could be called into hearings as swirling controversy threatens to derail the Republican firebrand's political ambitions.
The appointment has drawn generally favourable comments from Democrats and from some Republicans as well. A memo reportedly written by Comey was subsequently leaked, and it revealed that the president had asked Comey to end the investigation into former national security adviser Michael Flynn's ties to Russian Federation.
When asked about the Justice Department appointing Mueller as Special Counsel, the President deflected the question.
On Thursday, Mr Trump declared himself the victim of the "greatest witch hunt" in American political history and denied allegations of collusion.