"The answer, I think, is no for now", the California Republican said Sunday on "Fox & Friends".
Because Rosenstein oversees the Russian Federation probe, there was immediate speculation that the memos may have been leaked in order to damage Rosenstein, and by extension Mueller - more than to hurt Trump.
"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. Another official said that Rosenstein made the comments about taping Trump in jest.
The Times said Rosenstein told Justice Department and Federal Bureau of Investigation officials the secret recordings could be used to expose the chaos of the administration after revelations that Trump asked Comey to pledge loyalty to him and also divulged classified information to Russians in the Oval Office. Rosenstein has denied the reports of him discussing the 25th amendment.
Graham has previously said Trump could replace Attorney General Jeff Sessions but only after the midterm elections, a move that would allow Trump to install someone else to oversee the Mueller probe.
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.
Rosenstein branded the reports "inaccurate and factually incorrect".
"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. Some urged him to fire Rosenstein.
At the same time, at a rally in Springfield on behalf of Missouri Senate candidate Josh Hawley, Trump made a cryptic remark about removing the "lingering stench" from the Federal Bureau of Investigation and Justice Department but did not explicitly bring up the Rosenstein story.
"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", Schumer said.
The GOP is also concerned firing him may further jeopordise the confirmation of Judge Brett Kavanaugh to the Supreme Court which remains of a knife edge with news that Christine Blasey Ford, who has accused him of sexual assault, will testify before the Senate Judiciary Committee on Thursday.
"Rod deserves the right to be heard", said Rep. Trey Gowdy, R-S.C., on CBS' "Face the Nation".