That request came after a dinner at which Mr Trump is alleged to have asked asked Mr Comey to drop his probe into ex-national security adviser Mike Flynn the Times added.
Comey may once have tried to hide behind curtains to avoid Donald J. Trump, but the former FBI director's investigation could mean curtains for the Trump presidency.
The briefing came a day after Rosenstein appointed former FBI Director Robert Mueller as special counsel to investigate whether the Trump campaign coordinated with Russian Federation in that country's attempts to improperly influence the 2016 election.
I share in the confidence expressed by Democrats and Republicans alike that Mueller will do a fair and thorough job - and that FBI investigators, reportedly angry at the way Trump treated Comey, will look under every single rock.
Trump began the day by tweeting that he is the victim of "the single greatest witch hunt of a politician in American history", a claim ignoring impeachment efforts and blistering verbal attacks on previous presidents and other political leaders.
"I believe it hurts our country terribly, because it shows we're a divided, mixed-up, not-unified country".
President Trump took to Twitter to complain of a "witch hunt" Thursday, apparently in response to the appointment of a special counsel in the investigation of possible contacts between campaign aides and Russian Federation.
Democratic Senator Richard Blumenthal also told reporters that Mueller was conducting a criminal investigation, including into whether there was any obstruction of justice on behalf of Trump's campaign team.
"Some of them think it will pass, some are frustrated because they look like amateurs in responding to all of this, and some are freaked out because there is an actual investigation now", said one GOP operative in touch with White House officials.
It was widely reported that Russian Federation was hopeful of a resetting of the relationship between Moscow and Washington following Trump's election victory last November.
"Walls work. Just as Israel", Trump said, apparently referring to its West Bank barrier.
His reaction differed from Wednesday when he was quoted in a White House statement as saying: "I look forward to this matter concluding quickly".
"I hate to see anything that divides".
It's still early days in the Trump v. Comey heavyweight fight to define the reality of what exactly was said between the two men behind closed doors.
Rosenstein, the No 2 Justice Department official, named Mueller amid mounting pressure in Congress for an independent investigation beyond existing FBI and congressional probes into the Russian Federation issue.
The news that Rosenstein knew about Comey's firing before he wrote the memo could make the deputy attorney general a witness in Mueller's investigation, the Times noted.
Members of Trump's own party mostly backed his sacking of the Federal Bureau of Investigation chief, though some questioned its timing since Comey was investigating Trump's closest associates and possibly the president himself.
The FBI's reputation, he said, had been damaged during the presidential campaign, given "what happened with respect to the Clinton campaign". Tim Scott, R-S.C. "I do think that the special prosecutor provides a sense of calm and confidence perhaps for the American people, which is incredibly important".
"He had to throw some brushback pitches to the administration", Wittes said, according to The New York Times. Mr. Wittes said the conversations give an insight into the type of cozy relationship Mr. Trump wanted with Mr. Comey. "So, we want to get back and keep on the track that we're on, because the track that we're on is record-setting".