Trump, who said in a speech earlier on Wednesday that no politician in history "has been treated worse or more unfairly", has long bristled at the notion that Russian Federation played any role in his election victory. He did not offer any evidence of such acts in his reference to Democratic former President Barack Obama and former presidential candidate Hillary Clinton.
Meantime, President Trump lashed out on Twitter Thursday morning over the "witch hunt".
Mueller was appointed Federal Bureau of Investigation Director by former President George W. Bush in 2001 and served until 2013. Since Sessions has recused himself from the Trump Russia investigation, Rosenstein - who signed the letter appointing Mueller as "acting attorney general" - is the only person with the authority to curtail Mueller's work.
Trump's latest comments stood in contrast to his measured statement on Wednesday night after the announcement of Mueller's appointment, when he said "a thorough investigation will confirm what we already know - there was no collusion between my campaign and any foreign entity".
Trump says In a written statement that he looks forward to "this matter concluding quickly".
He also pledges to never stop fighting for the people and issues that are important to the country's future.
Mr Comey succeeded him, having been appointed by Mr Obama.
Multiple news outlets reported Tuesday that prior to ousting Comey, Trump asked him to terminate an investigation into former National Security Advisor Michael Flynn. He described Mueller as "credible" and "independent" and said his appointment would be "good for the Department of Justice". The goal of the special counsel probe, he said, was to determine whether there was collusion between Trump associates and Russian Federation to do so.More news: England's emergency units open 'as normal' again after cyber attack
Asked if Mueller's appointment would preempt his own request for Comey information, the Iowa Republican replied, "I don't know".
A special counsel is empowered to conduct the investigation independent of the Justice Department hierarchy, with a dedicated staff of his choosing. But GOP lawmakers have grown increasingly anxious since Trump fired Comey, who had been leading the bureau's probe - and after Comey associates said he had notes from a meeting in which Trump asked him to shut down the investigation into the Russian Federation ties of former National Security Adviser Michael Flynn.
Trump has consistently said Democrats are pushing the Russian Federation investigation and using it as an excuse to explain away their election loss in November. He did work for the Department of Justice for a dozen years as a US attorney, and was an assistant to Attorney General Dick Thornburgh under George H.W. Bush. No doubt this investigation should extend to the circumstances that led to the President's abrupt dismissal of James Comey, and to other critical matters that arise. "At the end of the day, we need a public accounting of what went on to restore faith in government", Grassley added.
Burr told reporters Thursday about the response from Flynn's lawyer.
Trump is still expected to hold a joint press conference with Colombian President Juan Manuel Santos on Thursday, but sources say he will not answer questions about the special counsel.
Deputy Attorney General Rod Rosenstein appointed Mueller on Wednesday to the position, reports CNN.
The No. 2 Senate Republican, John Cornyn of Texas, said he is supportive of investigations in Congress but expressed concern about the "proliferation" of hearings.
Trump fired back ahead of a briefing Thursday afternoon for all US senators being held by Deputy Attorney General Rod Rosenstein. Principal deputy White House press secretary Sarah Huckabee Sanders pointed to Rosenstein's memo as the reason Trump fired Comey, saying that Comey had committed "atrocities" in overseeing the FBI's investigation of Clinton.