Former FBI Director James Comey has agreed to testify before the Senate Intelligence Committee.
Sen. Mark Warner (D-VA), the vice chairman and ranking Democrat on the committee, said the ex-FBI head "deserves an opportunity to tell his story".
Lavrov met with Trump at the White House on May 10, the day after the President fired Comey.
Donald Trump told Russian officials that former Federal Bureau of Investigation chief James B. Comey was a "nut job", it has been reported. "I faced great pressure because of Russia", Trump reportedly said.
That flies in the face of the White House's public insistence that Comey's dismissal was not linked to his ongoing investigation.
Democratic Representative Alcee Hastings of Florida, told reporters that Rosenstein declined to answer a question in the House briefing about who directed him to write his memo or give a "timeline" of it.
The report cites two unnamed sources, one of whom read from a summary of Trump's meeting with the diplomats in the Oval Office.
During his meetings with lawmakers, Mr Rosenstein said that his conversations with Mr Sessions revealed his long-held belief that Mr Comey should be replaced, based on his public statements related to the investigation of former presidential candidate Hillary Clinton, beginning in July a year ago.
The Times said the official documents were based on notes taken from inside the Oval Office on May 10 - the day after Comey was sacked.
"I'm not under investigation", he added.
The week in a seemingly endless onslaught of news about chaos coming from within the White House shows no signs of slowing down.
Trump has called the furore since he fired Comey a "witch hunt" and insisted there was no contact between his campaign and Russian Federation.
Two more major and potentially damaging reports about President Donald Trump's dealings with Russian Federation were published Friday, shortly after the president departed for his first overseas trip.
The day after the conversation, The Times reports, White House chief of staff Reince Priebus had asked Comey to assist in deflecting media reports saying Trump's surrogates had been in contact with Russian intelligence agents before Trump's inauguration.
In a pair of morning tweets and at a later news conference, Trump described calls for his impeachment as "ridiculous" and said he had done nothing to warrant criminal charges.