Perhaps the most anticipated witness is former FBI Director Comey, who was sacked by Trump and reportedly wrote memos alleging the president pressured him to halt a probe of Michael Flynn, Trump's former national security adviser.
Russian Federation has denied United States intelligence agencies' conclusion that it interfered in the election campaign to try to tilt the vote in Trump's favor.
"We'll get rid of the smoke and see where the actual issues lie", said Sen. "I think we have a very divided country because of that and many other things. Zero", he said at a news conference with Colombian President Juan Manuel Santos.
But, in a Twitter rampage the day after, Mr. Trump said it was the single greatest witch hunt of a politician in American history.
"I'm fine with whatever people want to do", he added.
The move followed a week in which the White House was thrown into an uproar amid rising demands by Democrats and some of Trump's fellow Republicans for an independent probe of whether Russian Federation tried to sway the outcome of November's presidential election in favour of Trump and against Democrat Hillary Clinton.
Ryan said of the special counsel appointment of Robert Mueller was consistent with his goal of ensuring that "thorough and independent investigations are allowed to follow the facts wherever they may lead".
Rosenstein's decision to notify the White House only after making the pick goes a long way towards establishing the independence of both Mueller and the Justice Department.More news: Rosenstein Tells Senate He Knew of Comey Firing Before He Wrote Memo
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Sanders said on "This Week" that Congress should complete its review first, which she said she is confident will show that Trump associates had not involvement in Russia's actions.
And while it may be easier to ignore "the single greatest witch hunt of a politician in American history" as pure hyperbole, it can not be dismissed. Trump wrote, ignoring impeachment efforts and blistering verbal attacks on previous presidents and other political leaders. In the cases of Presidents Nixon, Reagan and Clinton, special prosecutors were conducting investigations that took place years into their presidencies.
Members of the Trump White House might need to lawyer up.
Mueller served as Federal Bureau of Investigation director for 12 years until Comey succeeded him in 2013.
He says he and Burr are planning to meet with Mueller early next week to talk about how to proceed.
Mr Mueller will have sweeping powers and the authority to prosecute any crimes he uncovers. The remarks were made shortly after the Senate met with Deputy Attorney General Rod Rosenstein.
Graham said that the nature of Mueller's investigation has hurt the Senate's ability to call witnesses in their counter-intelligence probe. The four congressional committees are largely counterintelligence investigations focused on what role Russia played in the 2016 presidential election, that alleged Russian interference. He was sacked as national security adviser in February after it emerged he had misled Vice President Mike Pence about conversations with Russia's ambassador to Washington in December. It was not clear what caused the mix-up.