Rosenstein, in a statement to NBC News, said, "The New York Times's story is inaccurate and factually incorrect."
"I will not further comment on a story based on anonymous sources who are obviously biased against the Department and are advancing their own personal agenda," Rosenstein said. "But let me be clear about this: based on my personal dealings with the President, there is no basis to invoke the 25th Amendment."
A senior Justice Department office who spoke with NBC News described a meeting on May 16, 2017, in a secure facility at the Justice Department, attended by Rosenstein, McCabe; FBI lawyer Lisa Page and four DOJ officials including Scott Schools, the department's senior career official who signed off on the decision to fire McCabe earlier this year.
The official said that during an argument at the meeting between Rosenstein and McCabe, Rosenstein asked, "Well, what do you want me to do, Andy, wear a wire?"
Two DOJ officials told NBC News that remark was made sarcastically.
The senior DOJ official said there is one very brief reference to the 25th Amendment in a note about the meeting, written later by McCabe: "DAG raises 25th Amendment." DAG refers to "deputy attorney general," Rosenstein's title. The official added that the notes about the same meeting taken by Page contain no reference to that amendment.
The official says Rosenstein did not raise the notion of invoking the 25th Amendment at the meeting.
McCabe's lawyer, Michael Bromwich, in a statement to NBC News said, "Andrew McCabe drafted memos to memorialize significant discussions he had with high level officials and preserved them so he would have an accurate, contemporaneous record of those discussions."
"When he was interviewed by the Special Counsel more than a year ago, he gave all of his memos -- classified and unclassified -- to the Special Counsel's office. A set of those memos remained at the FBI at the time of his departure in late January 2018. He has no knowledge of how any member of the media obtained those memos," Bromwich said.
The Department of Justice referred to that statement when contacted by NBC News and asked for comment about the The Times article.
The White House had no immediate comment when contacted by NBC News.
Trump's son Donald Trump Jr. referenced a famous line in the film "Casablanca" in a tweet about The Times story.
Laura Ingraham, the Fox News host, called on Trump to fire Rosenstein.
Rosenstein remained employed as of Friday afternoon.
Senate Minority Leader Chuck Schumer, D-NY, said in a statement, "This story must not be used as a pretext for the corrupt purpose of firing Deputy Attorney General Rosenstein in order install an official who will allow the president to interfere with the Special Counsel's investigation."
"Generals Kelly, Mattis and numerous other White House and cabinet officials have been reported to say critical things of the president without being fired," Schumer said.