And finally, on Wednesday, Rosenstein appointed a special counsel, former FBI Director Robert Mueller, to lead the investigation.
In a joint press conference with Colombia's President Juan Manuel Santos, President Trump said, "while I respect the move, the entire thing has been a witch hunt, I think it divides the country, I think we have a very divided country because of that and other things".
Rosenstein briefed USA senators on Thursday, but made no public comments.
Democrats and Trump's fellow Republicans widely praised Mueller's appointment.
Democrats - and many Republicans - had been eager to hear from Comey after reports emerged this week that Trump may have urged the then-FBI director to drop an investigation into former national security adviser Michael Flynn. Mueller will have sweeping powers and the authority to prosecute any crimes he uncovers.
"By having somebody like Bob Mueller head whatever investigation assures the American people that there's no undue influence, be it here, be it at the other end of Independence Avenue or within the Justice Department or FBI", said Burr, whose committee is conducting its own investigation. "That's a benefit to us", Lankford said. Flynn, through his lawyer, had earlier asked for immunity from "unfair prosecution" in exchange for agreeing to co-operate with the committee.
Mr Trump categorically denied he asked Mr Comey to drop his probe of Mr Flynn, whom he fired on Feb 14 for misleading Vice-President Mike Pence, about the extent of his conversations a year ago with Russia's ambassador.
Attorney General Jeff Sessions recused himself in March after acknowledging undisclosed contacts with the Russian ambassador during the campaign, leaving the matter in Rosenstein's hands.
"This was a very good first step".
Asked about the potential that there had been obstruction of justice, Republican House of Representatives Speaker Paul Ryan told reporters the special counsel will "follow the facts where ever they may lead" and that "it is premature to prejudge anything at this point".More news: Trump Rails Against 'Witch Hunt' Amid Special Counsel Appointment
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Trump's defiant comments stood in contrast to his measured statement on Wednesday night after former FBI chief Robert Mueller was named as special counsel, in which he said "a thorough investigation will confirm what we already know - there was no collusion between my campaign and any foreign entity".
"Look at the way I've been treated lately, especially by the media".
But he expressed more resentment Thursday in his tweets.
Rosenstein, who was appointed US attorney by President George W. Bush and held the job for the entire Obama administration, didn't immediately embrace the idea of a special counsel. He gave no examples. "I was stunned when President Trump suggested that he might have tapes of his meetings with director Comey, because those would become the subject of subpoenas".
In appointing Mueller, the Justice Department selected a seasoned law enforcement veteran who guided the Federal Bureau of Investigation through the aftermath of the September 11 attacks and led its terror-fighting efforts over the next decade. He was so valued that President Barack Obama asked him to stay on two years longer than his 10-year term. But Senate Democratic Leader Chuck Schumer said Comey should still appear before congressional committees that have invited him to testify.
The Justice Department said Mueller has resigned from his job at a private law firm, WilmerHale, to take the job.
"The political reason on the Democratic side is obvious: It's to namely keep this issue going and to keep it ideally going into the 2018 elections and, even more, into the presidential elections of 2020", he said.
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 Flynn's Russian Federation ties.
Trump is preparing to leave town Friday on his first foreign trip, and aides have been hopeful the journey will be a chance for the administration to get back on track after weeks of chaos and distractions.