In his Tuesday testimony before the Senate, Rosenstein said he would only fire Mueller under "lawful and appropriate orders" to prove Mueller's firing is for a "good cause".
Earlier, Rosenstein had authored an extensive and publicly-released memorandum recommending former FBI-chief James Comey's firing. Arkansas Gov. Mike Huckabee tweeted Thursday that Mueller called him to talk about the Russian investigation and he was relaying that information because he did not want the "NY Slimes" and "Wash Compost" to inaccurately report on that piece of news.
And Trump's apparent reference to Rosenstein, who oversees the Russian Federation investigation because attorney-general Jeff Sessions recused himself from it, came just hours after an oddly worded statement from Rosenstein complaining about leaks in the case. He has already been subpoenaed by the House Intelligence Committee.
Trump also compared his position to the investigation of Hillary Clinton's use of a private email server in another tweet.
At the December meeting with Kislyak, Kushner suggested establishing a secure communications line between Trump officials and the Kremlin at a Russian diplomatic facility, according to US officials who reviewed intelligence reports describing Kislyak's account.
Mr Rosenstein has been overseeing the Russian Federation probe since shortly after attorney general Jeff Sessions recused himself.
Rosenstein named Mueller as special counsel last month to lead the inquiry into Russia's meddling in last year's United States presidential campaign and whether anyone close to Trump colluded in that effort.
Including Trump's conduct in the investigation marked a major shift in the work of the investigators, and certainly a shift from what the FBI has been focused on for months, Russian interference in the 2016 elections and whether or not Trump operatives colluded with them. Trump put fresh pressure on the second-highest-ranking official at the Justice Department on Friday as he tweeted, "I am being investigated for firing the FBI Director by the man who told me to fire the FBI Director!"
Trump responded to reports he is personally being investigated for obstruction of justice with a characteristic scorched earth defense: claiming mistreatment of historic proportions and calling into question the probity of his accusers.
Trump allies have been questioning whether Mueller can lead an unbiased probe, given that some members of his team have made campaign contributions to Democratic candidates. It was unclear whether his tweet about being under investigation was based on direct knowledge or new media reports that suggest Mueller is examining whether the president obstructed justice by firing Comey.
Finally, we know that Mueller's investigators are also looking for "any evidence of possible financial crimes among Trump associates", as the Post phrased it.
Trump also attacked the media Friday morning.
Pence's decision to hire Richard Cullen, a former USA attorney, came less than a month after Trump hired his own private lawyer.
Several congressional panels are also investigating the Russian Federation issue, which has cast a shadow over the early months of Trump's presidency.
Trump has hired one of his longtime lawyers, Marc Kasowitz, to represent him in the multiple inquiries. Coats told associates that Trump asked him whether he could intervene with Comey to get the Federal Bureau of Investigation to back off its focus on Flynn, the officials said.
There is wide consensus that Trump is not doing himself any legal favors by repeatedly referring to it on Twitter.