Rosenstein told McCabe, who was also later fired by Trump, that he might be able to persuade Attorney General Jeff Sessions and John Kelly, the former homeland security secretary and current White House chief of staff, to invoke the 25th Amendment, the Times said.
According to The Times, none of Rosenstein's proposals were acted upon and it remains unclear the level of seriousness Rosenstein had when making the suggestions.
According to U.S. media, the source of the allegations are memos created by former Federal Bureau of Investigation acting director Andrew McCabe, who was sacked by Mr Trump in March this year.
U.S. President Donald Trump on Friday said documents from the federal Russian Federation probe would not immediately be released, just days after he ordered them to be made public, citing concerns by the U.S. Justice Department that doing so could harm the investigation.
Senate Democratic Leader Chuck Schumer said in a statement: "This story must not be used as a pretext for the corrupt objective of firing Deputy Attorney General Rosenstein in order to install an official who will allow the president to interfere with the Special Counsel's investigation".
Some analysts had suggested the Times story might have been planted by Trump allies seeking to justify a move against Rosenstein.
"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", he said in a statement.
McCabe has no knowledge of how the memos were made available, said his lawyer Michael Bromwich.
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It was victory that meant everything to Woods, who has been sidelined by a number of back surgeries over the last few years. Tiger Woods had huge galleries follow his final round as he looked to close out the Tour Championship golf tournament.
"Rod Rosenstein shld have been fired long ago for being part of the "resistance" and not providing documents to congress in order to save his corrupt pals", she wrote.
In his interview Pompeo, formerly the Central Intelligence Agency director, said he had "told it to junior folks at the Central Intelligence Agency, and the state department: we need everyone who's engaged in helping achieve President Trump's mission".
Despite unrelenting criticism from the White House on the course of the investigation into Russia's election interference, Rosenstein has offered unqualified support for Mueller. Mr Trump referred to the "Russia thing" when asked why he was removed.
A Justice Department official who met frequently with both McCabe and Rosenstein said that in the months that followed, Rosenstein never broached either subject - the 25th Amendment or a possible wiretap involving the president. That person's identity is unknown to journalists in the Times news department.
The New York Times reported Friday that Rosenstein floated the idea of using the 25th Amendment to remove Trump on grounds that he's unfit for office and suggested secretly recording Trump to expose chaos in the administration.
During Francisco's confirmation hearings past year, the Senate Judiciary Committee's top Democrat, Dianne Feinstein, mentioned his expressed support for Trump's travel ban and urged him in a letter to "publicly commit to refuse any order or request - whether express or implied - to interfere in the Special Counsel's investigation".
The announcement, in a pair of tweets, represented a highly unusual walk-back for a president who has pressed for the release of classified information that he believes will expose "really bad things" at the FBI and discredit special counsel Robert Mueller's investigation into possible coordination between Russian Federation and the Trump campaign.
Fox News host Laura Ingraham tweeted that Rosenstein "needs to go".
Deputy Attorney General Rod Rosenstein speaks during a news conference at the Department of Justice in Washington on July 13, 2018.