FBI Director James Comey was abruptly fired by the president on May 9, and Mueller was appointed special counsel by the Justice Department to continue an independent investigation. And Democrats and Republicans alike have criticized his use of Twitter, particularly amid the ongoing investigation into Russia's interference in the 2016 presidential election.
Virginia Democratic Sen. Mark Warner, the vice chairman, told the news network: "The criminal piece of the investigation will be handled by the special counsel, but if we find facts we can turn this over to the special counsel" and "report them" to Mueller's office.
The obstruction of justice investigation into Mr Trump began days after Mr Comey was sacked on May 9, according to people familiar with the matter, The Washington Post said. He is reportedly interviewing three top intelligence officials - Director of National Intelligence Dan Coats, National Security Agency Director Mike Rogers and retired deputy NSA director Richard Ledgett - as part of the probe as early as this week. Examining such possible charges will allow investigators to interview key people within the administration including Attorney General Jeff Sessions, Rosenstein and possibly Trump, the source said.
Mr Trump tweeted that the move was the latest action in a "phony story".
He tweeted early on Thursday morning, saying: "They made up a phony collusion with the Russians story, found zero proof, so now they go for obstruction of justice on the phony story".
Trump spoke to Coats and Rogers about the Russian Federation investigation, according to officials.
Trump also retweeted law professor Alan Dershowitz, who has called for a halt to conversations about obstruction of justice because there was "no plausible case". Trump said in a follow-up message.
Sessions vigorously denies improper Russian Federation contacts
And in the game of he-said, he-said between Trump and Comey, the president's history hands the victory to the G-man, hands down. He was obviously angry about some of the claims made by James Comey and the "secret innuendo" being thrown against him.
The tweet represented a remarkable show of frustration by a serving President, who is making no secret of his emotions as an investigation that started out as an effort to understand Russian meddling in the election appears to be moving ever closer towards his inner circle and narrowing in on his own conduct. "But it would probably be up to, probably, the US attorney for the District of Columbia to actually charge someone with contempt if Congress voted to hold them in contempt and obviously that person's boss is Jeff Sessions, so there's some potential conflicts there".
Special Counsel Robert Mueller is looking into Mr. Trump's words and conduct in the events of recent weeks, marking a stark shift in the investigation that initially was supposed to focus on Russian influence and any connections to the Trump campaign, but not necessarily Mr. Trump himself, the report said.
A spokesman for the office of the special counsel declined to comment, and so did a representative for the director of National Intelligence. "Nice", Trump wrote in his first 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.
Wednesday's Washington Post article caused a firestorm in the White House on Thursday, particularly because President Trump had been trying so hard to prove that he was not a focus of the Russian Federation investigation. When Comey testified last week, he also gave the committee and Mueller a document that involved Comey taking notes of his communications with President Trump. The first time he told Trump was in his first meeting with the president before the inauguration on January 6.
Marc Kasowitz, Trump's personal lawyer, said Comey's testimony vindicated that Trump wasn't under investigation by the FBI.