With Attorney General Jeff Sessions recused from investigations related to the 2016 presidential campaign, Rosenstein is the acting attorney general overseeing Mueller's probe.
If Mueller knew he was going to be named special counsel, it is unlikely he divulged that information to Trump.
After President Donald Trump's friend, Chris Ruddy, said Monday night that Trump is considering firing Mueller, Rosenstein said in a Senate appropriations subcommittee hearing that Mueller can "only be fired for good cause", and it would be his job to put that good cause into writing. The person spoke on condition of anonymity to discuss private conversations.
During testimony before members of Congress on Tuesday, Deputy Attorney General Rod Rosenstein refused to answer a question about who directed him to write the memo President Trump cited to justify the firing of former FBI Director James Comey, saying it might be part of Special Counsel Robert Mueller's investigation.
In his interview, Ruddy said Trump had interviewed Mueller for the position of Federal Bureau of Investigation director several days before Rosenstein appointed him special counsel.
Ryan echoed that sentiment, calling the reports about the president possibly moving to fire Mueller "a rumor".
Deputy Attorney General Rod Rosenstein says he has seen no evidence of good cause to fire the special prosecutor overseeing the Russian Federation investigation.
Anxiety about the probe - and fresh concerns about the political leanings of some of the attorneys involved - is percolating in the West Wing of the White House.
Mueller was interviewed both by the Justice Department and by the president personally for the director's position, according to a senior White House official. He said he would agree to dismiss Mueller only if there were a legitimate basis to do so.
Trump friend Chris Ruddy, the CEO of Newsmax, has gone so far as to suggest the president was already thinking about "terminating" Mueller.
"It is not the function of the Director to make such an announcement", Rosenstein wrote.
Spicer, the spokesman, declined to say then that Sessions enjoyed Trump's confidence, though spokeswoman Sarah Huckabee Sanders said later in the week that the president had confidence "in all of his Cabinet". Still, Gingrich said any special counsel with an agenda can "all of the sudden find something procedural and technical to latch onto".
Rep. Adam Schiff, the top Democrat on the House Intelligence committee, said any move to oust Mueller would "be the last straw" for many in Congress and would have "echoes of Watergate", when President Richard Nixon dismissed special prosecutor Archibald Cox over Cox's subpoenas for White House tapes.
Conservative commentator Ann Coulter offered a similar message, tweeting, "Now that we know TRUMP IS NOT UNDER INVESTIGATION, Sessions should take it back & fire Mueller". Jeanne Shaheen, a New Hampshire Democrat, whether he has seen good cause to fire Mueller. Gingrich said he is troubled by Democratic donations of Mueller's picks to help lead the probe.
"I think he's weighing that option", Ruddy said in an interview Monday with Judy Woodruff of "PBS NewsHour". "I don't question his integrity at all, but he has to be careful not to create the impression of partiality".