But Graham said that Mueller's appointment now shifts the FBI probe into a higher gear, and is likely to hamper the House and Senate intelligence committees' investigations into Russian electoral interference. The first tweet, about Hillary Clinton and former President Barack Obama's allegedly "illegal acts", was strangely deleted and reposted three hours later.
The president claimed that the Clinton campaign and Obama administration committed "illegal acts" but never received a special counsel.
Mr Mueller's broad mandate gives him not only oversight of the Russian Federation probe, but also "any matters that arose or may arise directly from the investigation".
That's according to several United States senators who met with Rosenstein Thursday afternoon in a secure room in the Capitol basement.
There used to be a process for the appointment of a much more empowered "independent counsel" - but Congress allowed that legislation to lapse in 1999. Congressional investigators have been seeking Comey's memos on his meeting with Trump, as well as documents from the Justice Department related to his firing. The White House counsel's office was alerted only after the order appointing Mueller was signed, said a senior White House official, who was not authorized to speak publicly by name and commented only on condition of anonymity.
A number of lawmakers exited the meeting stressing that Mueller's probe, while appropriately diving in to all possible Trump-Russia ties, may make it more hard for current congressional investigations to access vital intelligence.
The Justice Department announced Wednesday that former Federal Bureau of Investigation director Robert Mueller has been given sweeping power to investigate Russian interference in the 2016 presidential campaign, including potential links between Moscow and Trump campaign associates.
"I think we'll have a more hard time getting information here for the intelligence committee", Republican Senator Ron Johnson said.More news: Dozens of protesters demand Brazilian president's resignation
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Despite initially opposing appointment of an independent counsel, Republican House Speaker Paul Ryan said Thursday that the development "helps assure people and the Justice Department that they're going to go do their jobs independently and thoroughly, which is what we've called for all along".
Trump had been silent on Twitter about the most recent controversy in the wake of the bombshell reports that he asked former FBI Director James Comey to back off the investigation of former national security adviser Michael Flynn.
Trump categorically denied he asked Comey to drop his probe of Flynn, whom he fired on February 14 for misleading Vice President Mike Pence, about the extent of his conversations past year with Russia's ambassador.
The meeting, at which all senators were free to ask questions and make statements, had been expected to be contentious as anger flared over Comey's dismissal and Rosenstein's possible role in it.
The President's comments come after he angrily denounced the appointment of former FBI boss Robert Mueller to head the Russian Federation investigation in a series of angry tweets as "the single greatest witch hunt of a politician" in USA history.
Sen. Ben Cardin, a Maryland Democrat, said he also believes Rosenstein's memo had little to do with the president's decision. But Senate Democratic Leader Chuck Schumer said Comey should still appear before congressional committees that have invited him to testify.
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. "Congress clearly needs to investigate, and we need to hear from Comey now".