Ex-FBI boss James Comey, who was sacked by Donald Trump, has agreed to testify publicly before the Senate Intelligence Committee.
According to the U.S. daily, the quotes and other excerpts from the minutes was readout to its correspondent by a Trump administration official.
It said one official had read quotations to the Times and another had confirmed the broad outlines of the discussion.
"I hope that former Director Comey's testimony will help answer some of the questions that have arisen since Director Comey was so suddenly dismissed by the President", said Senator Mark Warren, Vice President of the Senate Intel Committee, in the press release.
The news broke shortly after Trump took off for his critically important five-country, eight-day foreign trip, the first of his presidency.
Shortly after the New York Times reported was published, Rep. Elijah Cummings, the ranking Democrat on the House Oversight Committee, asked the head of the committee, Republican Rep. Jason Chaffetz of Utah, to request the document mentioned in the story.
A White House official said Trump and Secretary of State Rex Tillerson would attend the signing of a memorandum of intent on a package of defence equipment and services to bolster the security of the kingdom and the Gulf region in the face of Iranian threats. "By grandstanding and politicizing the investigation into Russia's actions, James Comey created unnecessary pressure on our ability to engage and negotiate with Russian Federation". "It makes it very clear that what Donald Trump was trying to do was to end the Russian investigation". "Once again, the real story is that our national security has been undermined by the leaking of private and highly classified conversations".
The topics, including the Justice Department's decision to appoint Former FBI Director Robert Mueller as Special Counsel to investigate Russian meddling in the 2016 election, overshadowed President Trump's third visit of the week from a foreign leader. Whether or not Trump was trying to influence Comey and the FBI's investigation into the campaign's connections to Russian Federation will be at the center of the hearing to try to figure out whether the president obstructed justice.
And The New York Times reported that the president had told Russian officials he felt the dismissal of his Federal Bureau of Investigation director had relieved great pressure on him.
"There is no collusion between certainly myself and my campaign, but I can always speak for myself, and the Russians - zero", Trump told reporters. Despite knowledge of the probe, the incoming administration made a decision to go forward with plans to hire Flynn as the National Security Advisor, a position that gave him access to America's highest military and intelligence secrets. However the White House said there would be no announcement Friday.
"This matter is not going to distract the President or this administration from its work to bring back jobs and keep America safe", the official added. This renewed my confidence that we should not have confidence in this administration.
Trump met Russian Foreign Minister Sergei Lavrov and Russia's ambassador to Washington in the Oval Office the day after Trump fired Comey, who was in charge of the Russia election probe.
Rosenstein only told House members that he appointed a special counsel to restore Americans' faith in the Russian Federation investigation, saying that interference in US elections should not be a partisan issue.