When asked by a Sinclair Broadcast Group reporter if he made such a threat, Rosenstein responded, "No, I'm not quitting."
The Washington Post reported late Wednesday, citing a person close to the White House, that Rosenstein threatened to resign after White House press officials repeated a narrative which painted him as the key influence that led to Comey's dismissal.
The Trump administration has said Comey was terminated because of his handling of the FBI's investigation into Hillary Clinton's emails. The White House insists that Rosenstein and Attorney General Jeff Sessions initiated the conversation about Comey and that the president simply accepted their recommendation for his removal.
The Wall Street Journal reported Thursday, citing a person familiar with the conversation, that Rosenstein had urged White House counsel Don McGahn to amend the administration's portrayal of the process that led to Comey's firing.
The person told the Journal that the deputy attorney general implied he "couldn't work in an environment where facts weren't accurately reported."
The Justice Department declined to comment.
Rosenstein did sign the memo pinning Comey's termination on the ousted FBI director's conduct in the probe into Hillary Clinton's handling of classified information last year. White House officials have said the firing had nothing to do with the FBI's probe into Russian meddling in the 2016 election, which Comey first revealed publicly in March and includes any possible links between the Trump campaign and Russia.
Trump fired Comey on Tuesday, and the president said Wednesday that the FBI director "was not doing a good job." Comey reportedly found out that he had been fired around the time the news broke on television.
The White House claimed Wednesday that Trump's confidence in ousted FBI Director James Comey had eroded over the last "several months." Spokeswoman Sarah Huckabee Sanders said the president did not make the "final decision" to remove Comey until Tuesday — though he considered firing him since he took office in January.
Asked what changed recently about Trump's confidence in Comey, Sanders partly cited a correction that had to be issued to testimony that the FBI director gave in a congressional hearing last week. That was not included in the memo dated Tuesday in which Rosenstein outlined his recommendation to fire Comey.
The spokeswoman said that Sessions and Rosenstein came to Trump at a meeting on Monday with concerns about Comey. Trump then asked Rosenstein to give his recommendation in writing, Sanders said. She claimed that Trump had not decided to fire Comey before asking for the memo and did not make the decision until Tuesday.
On Tuesday night, The New York Times reported that Department of Justice officials had "been working on building a case against [Comey] since at least last week." Multiple reports Wednesday also said that Comey had recently requested more resources for the Russia probe.
Sanders said she was not aware if Trump knew about Comey's reported request.
—CNBC's Jacob Pramuk contributed to this report.