Rosenstein told a Senate appropriations subcommittee that Mueller reports to him and that he could only fire the special counsel for reasons set out in the special counsel regulation, including "good cause".
A White House official confirmed that president Trump interviewed Robert Mueller to replace James Comey as Federal Bureau of Investigation director just one day before he was named Special Counsel. So that is what I would do.
"The Attorney General may remove a Special Counsel for misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause, including violation of Departmental policies". And I think last night's report that he is directly investigating Trump for obstruction of justice does two things: Makes it more likely that Trump will decide to do it and increases the costs if he does.
Chris Ruddy, an ABC News contributor and a longtime friend of Trump and the CEO of the conservative media company Newsmax, first told PBS in an interview Monday that the president is "considering, perhaps, terminating the special counsel, he's weighing that option".
Ruddy's comments have received a lot of attention.
With few exceptions, Trump's allies are urging him not to move to dismiss Mueller, which would be both politically and technically complicated.
Spokeswoman Sarah Huckabee Sanders told reporters aboard Air Force One that "while the president has the right to" fire Robert Mueller, "he has no intention to do so".
Rosenstein's testimony comes amid public statements from Trump confidants that the American president is mulling removing Mueller from the investigation.
The White House downplayed Ruddy's comments.
Rosenstein has been overseeing the Russian Federation probe since shortly after Attorney General Jeff Sessions recused himself.
"Director Mueller is going to have the full degree of independence that he needs to conduct that investigation appropriately", Rosenstein said. Some of Trump's allies now claim the results Mueller comes up with in the Russian Federation investigation will be tainted and biased due to Mueller's relationship with Comey and Mueller's hiring of several attorneys to help him with the investigation that the Trump boosters deem to be partisan.
The White House, however, said Trump and Ruddy have never discussed the issue.
New Hampshire Sen. Jeanne Shaheen, a Democrat, opened her line of questioning, in the hearing that was supposed to focus on the Justice Department budget, by asking if Rosenstein had seen any evidence of good cause to fire Mueller.
"Their strategy is clear: they know or suspect the facts might not be good for the president, so they're trying to vilify the man who's in charge of finding them", the New York Democrat said.
Mueller's investigation will oversee an FBI probe into whether members of Trump's campaign team and his associates colluded to help Russian Federation with its interference in the election. And leading Republican legislators on Tuesday waved off the idea of firing Mueller, with House Speaker Paul Ryan saying that his advice was to "let Robert Mueller do his job" and Senate Majority Leader Mitch McConnell voicing confidence in the special counsel.