It was unclear whether Mr Trump's reference to "tapes" suggested there might be secret recordings of conversations that could be used to challenge any statements by Mr Comey, or whether it was simply a way of warding him off from commenting.
Trump's recent comments have also prompted fresh accusations of interference with the FBI's Russian Federation investigation.
The briefing began with National Security Adviser H.R. McMaster discussing the president's upcoming trip to the Middle East and Europe, despite threats from President Trump earlier in the day threatening to cancel press briefings in the face of fallout from Comey's dismissal.
Four Democratic senators are asking the Department of Justice's (DOJ) top watchdog to investigate any "possible political interference" in the FBI's investigation into Russia's meddling in last year's presidential election.
"He's getting beat up", Trump said. "It's all about the objective with which it's done".
"The unprecedented access that my committee and my staff has had, there would be no investigative team independently that you could set up that would have the access to the type of information we've had access to", Burr said.
But whether the unfolding controversy ultimately puts Trump's presidency at risk is more a question of politics than law.
The dinner took place just after the White House had been warned by the then attorney general that Mr Trump's national security advisor Michael Flynn had been "compromised" and was vulnerable to blackmail over ties to Russian Federation.
"No", Rosenstein said, adding he is "not quitting".
The dinner came in the early days of Trump's administration, as the FBI was investigating Russian meddling in the election and possible ties to Trump's campaign.
US President Donald Trump has warned fired Federal Bureau of Investigation chief James Comey against leaking material to the media.
"James Comey better hope that there are no "tapes" of our conversations before he starts leaking to the press", Trump tweeted.
This was contradicted by Mr Trump on Thursday, when he told NBC News he had planned to fire Mr Comey regardless of Mr Rosenstein's recommendation.
That led Trump to tweet Friday morning, "As a very active President with lots of things happening, it is not possible for my surrogates to stand at podium with ideal accuracy".
The issue is not Trump's legal authority to dismiss Comey, which he possesses with or without cause.
"Just don't have them", he said after pausing. "I've talked to the president. Did he order Comey to end the investigation?"
The questions about Trump's motives are being fueled, increasingly, by his own words in interviews and on Twitter.
The Constitution specifies that the president can be removed only for treason, bribery, or "other high Crimes and Misdemeanors", the definition of which it leaves to Congress. Wording of this one had some TV news talking heads talking Richard Nixon and Watergate again.
For the first time since an Oval Office taping system was removed by President Richard Nixon's chief of staff almost 44 years ago, a president has hinted that White House conversations might again be secretly recorded.
Like many administrations, the relationship between Trump and the press has been tense.
She left after about an hour and a half inside the building and declined to comment to reporters.
He added: "It is my belief that the FBI will continue to pursue this investigation vigorously and completely".
After a meeting with Senate intelligence committee leaders on Capitol Hill Thursday, Rosenstein was asked by a reporter whether he threatened to quit. If so, President Donald Trump is following a problematic precedent.