Rosenstein is under an intense spotlight - and fire from Trump - as he oversees the probe by Special Counsel Robert Mueller into whether the real estate mogul's campaign colluded with Russian Federation to influence the 2016 election.
"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 added.
According to reports, senior Republicans have convinced a furious Mr Trump not to axe Mr Rosenstein, at least until the midterms are over for fear of negative blowback from voters.
Attorney General Jeff Sessions, Supreme Court nominee Brett Kavanaugh and Hurricane Florence were the top names in news headlines this past week.
The reported conversation about possibly secretly recording the president took place at a tense May 2017 meeting during the tumultuous period that followed Trump's firing of FBI Director James Comey. If Rosenstein's involved, he should be fired.More news: Destiny 2 Breakthrough mode arrives next week with new maps
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"We believe our first duty is to this country, and the president continues to act in a manner that is detrimental to the health of our republic", the writer said.
Trump also criticized Sessions' handling of immigration, saying he was "disappointed" with the attorney general for "many reasons".
Trey Gowdy, chairman of the House Oversight Committee, said Rosenstein should get a chance to explain his side of what happened after a review of evidence, including memos written by McCabe.
Democrats have warned Trump not to use the Times report as a pretext for firing Rosenstein and interfering with the Mueller probe.
"I'm not going to comment on that in any way other than to say this: I've been pretty clear since my beginning of service here in this administration, if you can't be on the team, if you're not supporting this mission, then maybe you ought to find something else to do", Secretary of State Mike Pompeo told "Fox News Sunday".