In a statement he and Republican Mike Conaway said they were seeking the documents as part of their own ongoing probe into alleged Russian interference into the 2016 U.S. election.
Rosenstein, the No. 2 Justice Department official, named Mueller amid mounting pressure in Congress for an independent investigation beyond existing FBI and congressional probes into the Russian Federation issue.
Mr Mueller's appointment has been welcomed by politicians from both sides.
Trump later said he meant to fire Comey regardless of the Justice Department's advice, and that he had the FBI's probe into alleged Russian election interference in mind when he did so.
"It's special. The FBI has not had that special reputation with what happened in the campaign, what happened with respect to the Clinton campaign, and directly and indirectly to the more successful Trump campaign".
He later reposted the same tweet, after correcting his misspelling of the word "counsel".
"There is no collusion between certainly myself and my campaign - but I can only speak for myself - and the Russians".
The meeting came after a week of White House chaos in which Trump also was accused of divulging classified information during a White House visit by Russia's foreign minister and ambassador to the United States. Claire McCaskill, after a Senate meeting with Rosenstein, said Rosenstein was aware Comey would be fired before writing the memo justifying Comey's firing.
McCaskill and other Senators were briefed by Rosenstein about his memo which was initially credited by the White House for the firing of former FBI Director James Comey. "And if Congress is going to spend so much time and energy looking into the Trump administration, how much time do they have for taxes and health care"? The new FBI director will face confirmation hearings, which will provide a fresh opportunity to maintain attention on the Russian Federation investigation. If Democrats take control of the House after the midterm elections next year, Trump is in far more trouble.
At a White House lunch with reporters on Thursday, Trump was asked about the appointment of a special counsel.
Mr Trump appointed Mr Flynn as his national security adviser weeks later despite the warning, but he was sacked after just 24 days.
"If I were Mr. Mueller I would jealously guard the witness pool", Graham said. In addition, Comey wrote a memo detailing Trump's comments to him in February saying "I hope you can let this go", referring to the Flynn probe, a source who saw the memo said on Tuesday.
"The takeaway I have is that everything he said was that you need to treat this investigation as if it may be a criminal investigation", said Sen.
Lashing out after a high-powered special counsel was appointed to investigate events around the 2016 election, Trump denied he ever plotted with the Kremlin to win the White House. The nation has never had an administration so embattled so early in its term.
According to associates of Trump, the president has become mistrustful of many around him amid a spate of leaks about West Wing dysfunction. News reports have characterized the president as dissatisfied with his communications team's inability to control the tenor of the coverage of recent negative stories.
Comey made it clear that he wouldn't tell the committee about his interactions with the White House. He understands pressure-cooker politics and knows how to navigate the corridors of power.
Mueller will have almost unfettered access to witnesses and information, and the ability to bring criminal charges.
The Senate intelligence committee also asked acting FBI Director Andrew McCabe to turn over any notes Comey might have made regarding discussions he had with White House or Justice Department officials about Russia's efforts to influence the election.
Independent investigations often take on a life of their own and can reach unexpected conclusions.