Mueller is a former federal prosecutor for the Justice Department and was appointed as FBI Director days before September 11, 2001 under President Bush. Trump wrote, referring to former President Barack Obama and former presidential candidate Hillary Clinton.
"With all of the illegal acts that took place in the Clinton campaign & Obama Administration, there was never a special councel appointed", Trump said. He made the same comments to reporters at the White House alongside Colombia's President Juan Manuel Santos, again insisting there was no collusion between his campaign and Moscow.
Mueller was named as special counsel on Wednesday, one day after a report surfaced that Trump had requested ex-FBI Director James Comey end any investigations involving National Security Adviser Michael Flynn's contacts with Russian Federation.
Senators said that Rosenstein steered clear of specifics while making clear that Mueller has wide latitude to pursue the investigation wherever it leads, including potentially criminal charges.
Rosenstein was due to brief US senators later on Thursday.
A key issue Mueller may have to tackle is whether Trump has committed obstruction of justice, an offense that could be used in any effort in the Republican-led Congress to impeach him and remove him from office.
The motivation and timing of Rosenstein's three-page memo has been scrutinized because the White House offered inconsistent explanations for it.
Republicans have largely stood behind Trump as the FBI and congressional investigations into Russia's election meddling intensified.
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Asked point-blank if he'd done anything that might merit prosecution or even impeachment, he said no and then added concerning the allegations and questions that have mounted as he nears the four-month mark of his presidency: "I think it's totally ridiculous".
As for the Russian Federation investigation being the greatest "witch hunt of a politician" in this country's history, it's apparent that Trump needs a history lesson - or to even look a few months back to the presidential election. Stocks and the dollar tanked, amid the whirlwind of Washington turmoil, as Wall Street rightfully anxious that Trump won't be able to pass his economic growth agenda of tax reform and deregulation - because Congress is busy chasing down the latest on the alleged Russian Federation conspiracy.
"I think that's an open question", said Democratic Senator Chris Coons of DE when asked whether Comey would appear before Congress, adding that it remained to be seen "how ongoing congressional investigations will be coordinated with the special counsel".
The tweets also raise questions about whether Trump, with his more defiant tone on the special counsel, will actually cooperate with the probe.
Senate Majority Leader Mitch McConnell says Robert Mueller's appointment as special counsel shows the investigation into Russian meddling in last year's USA election "will continue".
As he said, taking a shot at the media who is assuredly against him as a group, you have to fight and never give up.
The chairman of the Senate intelligence committee says his panel has not received a response from Michael Flynn's lawyer - correcting his earlier statement that Flynn would not comply with a subpoena.
His argument: Congressional interviews and hearings could inadvertently harm or complicate the criminal investigation.