Trump, however, tweeted he fired FBI Director James Comey, which triggered the obstruction of justice probe, based on the recommendation of Mueller and Rosenstein. A day later, Mr Rosenstein appointed Mr Mueller, who was at the helm of the Federal Bureau of Investigation during the September 11 attacks, as special counsel. Rosenstein will now have to do so - soon.
At a ceremony last month to welcome Associate Attorney General Rachel Brand, the Justice Department's third-ranking official, Rosenstein joked awkwardly about being at the center of criticism since taking office, according to people who were in the room.
Rosenstein also has authority to fire Mueller, if he finds evidence of good cause.
Earlier this week USA media reported that Mr Mueller was investigating the president for possible obstruction of justice. In the interview Trump did with NBC's Lester Holt shortly after firing FBI Director James Comey, Trump described Rosenstein's involvement in the firing.
"Mueller and Rod Rosenstein?"
"His concern was, because we're looking at the potential - again, that's the subject of the investigation - coordination between the campaign and Russian Federation, because it was President Trump - President-elect Trump's campaign, this person's view was, inevitably, his behavior, his conduct will fall within the scope of that work", Comey said.
Rosenstein could have asserted that although the regulations allow the attorney general (here Rosenstein because of Jeff Sessions's recusal) to ask for explanations of the special counsel's investigative process, and require periodic reports from the special counsel to the attorney general, he wasn't going to be supervising Mueller's investigation.
Yet Trump's angry tweets on Friday underscored the near-impossible challenge his advisers and legal team have in trying to get him to avoid weighing in on an active probe.
Trump fired Comey on May 9, but later in an interview on NBC said that the Russian Federation probe was on his mind when he made the decision to fire Comey. Former House Speaker Newt Gingrich this week called Mueller "now clearly the tip of the deep state spear aimed at destroying or at a minimum undermining and crippling the Trump presidency".
So Rosenstein will have to recuse himself. And allies of former President Bill Clinton also raised concerns of political bias by Kenneth Starr, who led the Whitewater investigation that ultimately led to Clinton's impeachment.
There were several unconfirmed press reports Friday, citing unnamed sources in the Justice Department, that Rosenstein was preparing to recuse himself. She was appointed to the Privacy and Civil Liberties Oversight board in 2012 by President Obama, then appointed to the associate attorney general position by Trump.
In short, Brand's a first-class conservative Washington lawyer, but not a prosecutor. Trump's only recourse is to dismiss Mueller outright.
The point is that Mueller's investigation is broadening, not narrowing. CNN also reported last month that Kushner may have tried to create a secret channel of communications between Trump's transition team and top Russian officials, including Kislyak.