His offer added a weird twist to the furor over Trump's intelligence disclosures.
Mr Comey's dismissal sent shockwaves through Washington, with critics accusing the president of trying to thwart the FBI investigation into Russia's alleged interference in the USA election and any ties to Trump associates.
"I hope you can let this go", Trump told Comey, according to a source familiar with the contents of the memo.
But reports claim Trump divulged an element of the intelligence - related to an ISIS plot against air travel - that could have allowed Russian Federation to divine the sources and methods used to gather the information.
A breach of trust raises the possibility that US friends might curtail such intelligence partnerships out of concern their secrets - and their sources and methods - could end up in the wrong hands.
At one point, Putin jokingly said he needed to upbraid Lavrov for failing to share the "secrets" with him.
Putin spoke at a time when in the USA, talk of impeaching Trump has migrated from the fringes of political discourse to the mainstream, as NPR's Domenico Montanaro reports.
"The idea that we would accept any evidence from President Putin is absurd", she said, smiling.
The New York Times said that during the Oval Office meeting, Trump condemned a series of government leaks to the news media and said the Federal Bureau of Investigation director should consider prosecuting reporters for publishing classified information.
Mike Schmidt, reporter for the New York Times, talks with Rachel Maddow about his scoop that James Comey's notes from a meeting with Donald Trump include a description of Trump asking Comey to end the investigation into disgraced Trump NSA Mike Flynn.
In Washington, Republican and Democratic lawmakers said they wanted to see the Comey memo.
House Oversight Committee chair Jason Chaffetz demanded that all correspondence be presented by 24 May. "That's all I really have to say about him".
Comey's memos could be valuable in any obstruction investigation.
"We have the White House saying that the conversation didn't occur". The Illinois senator said he wants to see Comey's memos and hear his testimony in public.
The White House vigorously denied it all.
"What else can these people can come up with, people who generate this kind of nonsense, this kind of delirium?" he asked.
Trump fired Flynn on February 13, on grounds that he had misled Vice President Mike Pence and other officials about his contacts with Russians.
Sen. Patrick Leahy (D-Vermont) tweeted out, "Memos, tapes, testimony, appt [sic] of special prosecutor - all are needed ASAP to get to the bottom of this cascading crisis". "It's a matter of national security, so I'm giving it to the @nytimes instead of the DOJ", one Twitter user elaborated.
"It was nothing that you would not know from open-source reporting", National Security Adviser H.R. McMaster said on Tuesday.
President Putin deployed his trademark sarcasm on Wednesday to make clear he thought the accusation that Trump had divulged secrets absurd.
Associated Press writers Vivian Salama and Jill Colvin contributed.
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