Sen. Mark Warner, D-Virginia, and ranking member of the committee, has said he wants Flynn to testify before the committee, a move that Sen. Richard Burr, R-N.C., the top Republican, said could happen "eventually."
Both members have said they would like to see the transcripts of the calls between Flynn and Russian Ambassador Sergey Kislyak.
The House Intelligence Committee
The House Intelligence Committee is less bullish about its investigation than its counterpart in the Senate.
While it is investigating Russian interference in the election, Chairman Devin Nunes, R-Calif., is a close ally of Trump's and has been lukewarm about an aggressive probe into Flynn. Nunes said that the ongoing investigation could be expanded to include Flynn if "it all falls under the umbrella."
House Democratic Leader Nancy Pelosi is skeptical of House Republican commitment to investigate.
"They are stonewalling this," Pelosi said. "The speaker is saying it's up to the Intelligence Committee — the chairman of the Intelligence Committee is saying don't look at me, I'm not doing any of this. the American people deserve better."
Like President Trump, congressional Republicans have expressed concerns about the leaks of intelligence to the media regarding Flynn and his phone call with Kislyak. While Nunes, chair of the House Intelligence Committee, has said that those who leak "belong in jail," he has not yet committed to investigating them.
House Speaker Paul Ryan, however, said the House Intelligence Committee should look into it.
"What I do worry about, though, is if classified information is being leaked. That is criminal," Ryan said. And so I think there should be an investigation as to the leaks of information leaving, wherever they're coming from."
Trump has focused on the leaks, saying that the leaks are more scandalous than the Flynn controversy.
The top Republicans of the House Oversight and Government Reform Committee and the Judiciary Committee sent a letter to the Department of Justice inspector general regarding "potential inadequate protection of classified information."
"We request that your office begin an immediate investigation into whether classified information was mishandled here," the letter said.
And the Senate Intelligence Committee is reluctant to open a probe into leaks. Burr said that leaking should be investigating by the FBI because of the criminal component to it, adding that the Intelligence committee doesn't have prosecutorial authority.
Russian Payment to Flynn
In the House Oversight and Government Reform Committee, Republican Chairman Jason Chaffetz and Ranking Member Democrat Elijah Cummings sent a joint letter to the Department of Defense asking about payments Flynn received from the Russian government for a trip in 2015.
"We are attempting to determine the amount Lieutenant General Flynn received for his appearance, the source of the funding, and whether he may have received payments from any other foreign sources for additional engagements," they wrote.
The top Democrat and Republican on the Senate Judiciary Committee have jointly written a letter asking that the FBI brief them on the circumstances leading up to Flynn's resignation.
While the Judiciary Committee does not deal with classified material, both Sens. Dianne Feinstein, D-Calif., and Chuck Grassley, R-Iowa, are interested in preservation of documents and to be able to see unclassified version of related materials, potentially opening another investigation from this committee.
While it's not bipartisan, in the House, Democrats are also calling on the Director of National Intelligence to brief them. House Minority Leader Nancy Pelosi and Rep. Adam Schif, top Democrat on the House Intelligence Committee, said in a letter that they'd like an "immediate briefing on the counterintelligence implications of these alarming actions."