UPDATE 2: Rosenstein issues another denial:
UPDATE: From The Hill:
The House Judiciary Committee announced on Friday that it intends to subpoena memos from former acting FBI Director Andrew McCabe detailing reported comments made by Deputy Attorney General Rod Rosenstein in which he proposed secretly taping conversations with President Trump and initiating a process to remove the president by invoking the 25th Amendment.
“I intend to subpoena “McCabe Memos” & all other docs that have been requested & not provided,” the committee tweeted Friday evening from its verified Twitter account.
House Judiciary on NY Times article: I intend to subpoena 'McCabe Memos' (The Hill)
The New York Times reports Deputy Attorney General Rod Rosenstein, in 2017 just after President Trump fired FBI Director James Comey, mentioned the possibility both of secretly recording the president and recruiting cabinet members to invoke the 25th Amendment which could be used to remove Trump from office.
Mr. Rosenstein made these suggestions in the spring of 2017 when Mr. Trump’s firing of James B. Comey as F.B.I. director plunged the White House into turmoil. Over the ensuing days, the president divulged classified intelligence to Russians in the Oval Office, and revelations emerged that Mr. Trump had asked Mr. Comey to pledge loyalty and end an investigation into a senior aide.
Mr. Rosenstein was just two weeks into his job. He had begun overseeing the Russia investigation and played a key role in the president’s dismissal of Mr. Comey by writing a memo critical of his handling of the Hillary Clinton email investigation. But Mr. Rosenstein was caught off guard when Mr. Trump cited the memo in the firing, and he began telling people that he feared he had been used.
Mr. Rosenstein made the remarks about secretly recording Mr. Trump and about the 25th Amendment in meetings and conversations with other Justice Department and F.B.I. officials. Several people described the episodes, insisting on anonymity to discuss internal deliberations. The people were briefed either on the events themselves or on memos written by F.B.I. officials, including Andrew G. McCabe, then the acting bureau director, that documented Mr. Rosenstein’s actions and comments.
Rosenstein says the NYT's reporting is wrong and pushing an agenda.
CNN:
"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 in a statement Friday obtained by CNN. "But let me be clear about this: based on my personal dealings with the President, there is no basis to invoke the 25th Amendment."
(...)
McCabe's lawyer, Michael Bromwich, said in a statement to CNN that his client "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 added.
A source who was in the room told CNN that the wire comment was "sarcastic and was never discussed with any intention of recording a conversation with the President."
Other follow-up reporting corroborates the understanding that Rosenstein made the "wire" remark sarcastically and offers conflicting evidence of whether he actually brought up the 25th Amendment.
Deputy Attorney General Rod Rosenstein was joking when he discussed wearing a wire to secretly record President Donald Trump and does not believe Trump should be removed from office through the use of procedures outlined in the Constitution's 25th Amendment, according to Justice Department officials who requested anonymity to discuss the conversation.
(...)
In a May, 16, 2017 meeting at a secure facility at the Justice Department — one week after Trump fired FBI Director James Comey — Rosenstein was arguing with Andrew McCabe, then the acting director of the FBI, about the president, according to a senior Justice Department official.
"Well, what do you want me to do, Andy, wear a wire?" Rosenstein asked at the meeting, which also included FBI lawyer Lisa Page and four career DOJ officials, according to the senior official. One of the career civil servants was Scott Schools, who would later go on to sign off on the firing of McCabe, the official said.
This official and a source who was in the room characterized Rosenstein's remark as sarcastic.
The senior official further said that the reference to invoking the Constitution to remove Trump comes from a post-meeting memo written by McCabe that said the deputy attorney general "raises 25th amendment" and that Page's notes from the same meeting do not contain any similar note.
While McCabe’s memos assert both the recording and 25th amendment conversations occurred at a meeting within days of Comey’s firing, another person at the meeting, speaking on condition of anonymity to discuss internal deliberations, insisted the recording comment was said in a moment of sarcasm, and that the 25th amendment was not discussed.
That person said the wire comment came in response to McCabe’s own pushing for the Justice Department to open an investigation into the president. To that, Rosenstein responded with what this person described as a sarcastic comment along the lines of, “What do you want to do, Andy, wire the president?”
That person insisted the statement was never discussed with any intention of recording a conversation with the president.
Another official at the meeting, then-FBI lawyer Lisa Page, wrote her own memo of the discussion which does not mention any talk of the 25th amendment, according to a second person who was familiar with her account.
A third person familiar with the discussions said McCabe had privately asserted previously that Rosenstein suggested invoking the 25th amendment and the idea of a senior law enforcement officials wearing a wire while talking to Trump.
President Trump previously has spoken about McCabe's memos, calling them lies and working to discredit them. It would be a sharp change of direction now should he use a description of what they allegedly contain as a reason to fire Rosenstein.
CNN:
CNN and others previously reported that McCabe has turned over to Mueller his contemporaneous notes on what Comey told McCabe about his private interactions with Trump, McCabe's own interactions with Trump and McCabe's impressions of meetings with Rosenstein.
Not long after those reports, Trump said McCabe "never took notes" in meetings and suggested that his memos were "fake."
"Spent very little time with Andrew McCabe, but he never took notes when he was with me," Trump tweeted. "I don't believe he made memos except to help his own agenda, probably at a later date. Same with lying James Comey. Can we call them Fake Memos?"
Trump has also painted McCabe as a serial liar. In April, after the Justice Department inspector general released a report damaging to McCabe, Trump promoted the report on Twitter and said it was a "total disaster" for McCabe. Trump added, "He LIED! LIED! LIED!"
At the same time, Trump and his congressional allies have been fiercely critical of Rosenstein, and the New York Times account could give the president the cover he wants to fire the man who oversees the Russia investigation.
"I am being investigated for firing the FBI Director by the man who told me to fire the FBI Director!" Trump tweeted in June 2017. "Witch Hunt."
This summer, two top Trump allies, Republican Reps. Mark Meadows of North Carolina and Jim Jordan of Ohio, introduced articles of impeachment against Rosenstein over his handling of House GOP inquiries into the use of a special federal court that ordered the wiretapping of Trump campaign adviser Carter Page, Republican requests for internal Justice Department and FBI documents and his own involvement in related matters that they say amount to a "conflict of interest."
That effort, which was opposed by House Speaker Paul Ryan, R-Wis., died over the summer, in part because House Republicans realized that a hypothetical Senate removal proceeding could interfere with that chamber's effort to confirm a new Supreme Court Justice.
Just today, President Trump backtracked on what Jordan and Meadows also had wanted – exposing unredacted pages of the Page FISA warrant – but firing Rosenstein for what the Times reports would be delivering his allies exactly what they've requested without having to disclose classified information.
Rosenstein Suggested Secretly Recording Trump and Discussed 25th Amendment (NYT)
McCabe memos say Rosenstein considered secretly recording Trump (WaPo)
New York Times: Rod Rosenstein discussed secretly taping Trump (CNN)
Rosenstein joked about secretly recording Trump, Justice Department officials say (NBC News)