White House officials have cited Rosenstein's memo as one of the main justifications for Trump's firing of Comey.
The FBI's investigation into the Trump campaign's ties with Russian Federation has extended to a current senior White House official, the Washington Post reported Friday.
The president's first stop is Saudi Arabia, where he will address a summit of Arab and Muslim leaders.
Comey is expected to speak before the committee after May 29 - the end of Congress's Memorial Day break.
The ex-FBI chief has not spoken publicly since his surprise firing last week. "Moreover, the American people deserve an opportunity to hear it", Warner said. Trump aides said he never tried to squelch the Flynn investigation nor made inappropriate disclosures to the Russians.
"We're extremely disappointed in James Comey's decision not to testify voluntarily before the Judiciary Committee", said Chairman Chuck Grassley and the panel's top Democrat, Dianne Feinstein, urging Comey to reconsider. "Given his commitment to the people and the mission of the Federal Bureau of Investigation, we expected him to be responsive to the senators responsible for vetting its next proposed leader".
Democrats have argued that Mr Trump's firing of Mr Comey during his Russian Federation investigation could amount to obstruction of justice.
Trump on Thursday lashed out at the appointment of a special counsel to investigate allegations that his campaign collaborated with Russian Federation to sway the 2016 election, tweeting that it was "the single greatest witch hunt of a politician in American history!"
"Did you at any time urge former FBI Director James Comey, in any way, shape or form, to close or to back down the investigation into Michael Flynn".
Air Force One departed from Joint Base Andrews, Maryland, outside Washington, with the president, first lady Melania Trump, his daughter Ivanka and son-in-law Jared Kushner, as well as a group of senior advisers and journalists covering the trip.
"I just fired the head of the F.B.I. He was insane, a real nut job", Trump said, according to a document summarizing the meeting that a USA official read to the Times.
Burr says the committee wants to hear from Comey on his role in the development of the USA intelligence agencies' assessment that Russian Federation interfered in last year's election. His testimony in front of the House of Representative may raise more questions than it answers, though, because it seems to contradict what the White House and attorney general Jeff Sessions have already said on the subject.
The developments were a blow to White House efforts to tamp down interest in the Russian Federation investigation as Trump and his staff boarded Air Force One for Saudi Arabia, first stop on his first foreign trip as president.
Intercepted Russian communications suggest that Russian officials felt that they had built up such a strong relationship with Flynn that they could use him to influence Trump and his inner circle, CNN reported.
The Russian government is accused by USA intelligence of trying to tilt the 2016 election in Trump's favor.
Did he or didn't he?
Two more bombshell reports published Friday afternoon delivered a one-two punch to the Trump White House regarding the investigation into its potential ties to Russian Federation, capping off a week where each day dealt a fresh blockbuster blow to the administration. House Democrats demanded an investigation into Trump immediately after the memo was released.