This would have been in the tumultuous days after James Comey was sacked as Federal Bureau of Investigation director, with the president citing in part a memo penned by Rosenstein - reportedly catching him off guard.
And Rosenstein may not be out of the woods with the White House: the president has already spoken to another official about becoming the acting deputy attorney general, sources told ABC News, and Rosenstein is expected to meet Trump in person on Thursday "to discuss the recent news stories", White House press secretary Sarah Sanders said in a statement.
"I never pursued or authorized recording the President and any suggestion that I have ever advocated for the removal of the President is absolutely false", he said in a follow-up statement more forcefully denying the New York Times story.
The Rosenstein furore came just six weeks ahead of the November 6 congressional elections, and his removal could become an explosive political issue as Trump's fellow Republicans try to keep control of Congress.
Trump was noncommittal about the possibility of firing Rosenstein.
Trump also said Rosenstein, a former US attorney, was "hired by Jeff Sessions".
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The removal of Rosenstein could became another flashpoint in the midterm congressional races, as Republicans are seeking to prevent the Democrats from gaining control of the House - where the impeachment process could be initiated - and possibly the Senate after November 6.
The Rosenstein furore, kicked off by unconfirmed reports that he had verbally resigned, underscored the mounting tension in the White House over the investigation by special counsel Robert Mueller into Russia's role in the 2016 presidential election.
Trump announced the nomination of Rosenstein to be deputy attorney general in January 2017.
The White House delayed until at least Thursday a decision on the fate of Rod Rosenstein, the Justice Department official overseeing the Trump-Russia investigation, following chaotic hours of breathless and sometimes conflicting reports anticipating his imminent departure. Trump and his supporters argue the Justice Department has run amok with "bias" and abuse of power, which they say includes inappropriate snooping on Trump's campaign in 2016.
U.S. Treasury yields fell as much as 2 basis points after the Axios report, signalling investor concern but later pared losses.
Some news organizations reported that Rosenstein had submitted a verbal resignation to White House officials.
Mr Trump has regularly dismissed the investigation as a "witch hunt".
Those developments came one week after Rosenstein laid the groundwork for the firing of Comey by writing a memo that criticized Comey's handling of the FBI investigation into Hillary Clinton's email server. Trump has repeatedly attacked Sessions for that move. In an interview with Geraldo Rivera that aired Monday, Trump laid blame for the controversy at the feet of his attorney general.