The Intelligence committee's vice chairman, Virginia Democrat Mark Warner, said the panel also had issued subpoenas to two Flynn-related businesses in Virginia for documents in the probe, and that one had already been served.
Responding to the prospect that Gen Flynn would refuse to turn over documents, Senator Dianne Feinstein, the top Democrat on the Senate Judiciary Committee, said in an emailed statement that "both the Intelligence and Judiciary Committees should continue to seek other ways to gain access to this information".
"The context in which the committee has called for General Flynn's testimonial production of documents makes it clear that he has more than a reasonable apprehension that any testimony he provides could be used against him", Kelner said.
Flynn has been at the center of the allegations of Trump campaign collusion with Russian efforts to interfere in last year's presidential elections.
His legal representative has previously demanded immunity from "unfair prosecution" before his client testifies.
The Fifth Amendment gives an individual the right to avoid self-incrimination.
The refusal to comply risked Flynn being cited for contempt of Congress.
"General Flynn's claims directly contradict documents the Committee has obtained showing that RT paid more than US$45,000 for General Flynn's participation in the 2015 gala in Moscow", Mr Cummings said in the letter.
Cummings, however, only widened the accusations against the retired lieutenant general in a letter addressed to Jason Chaffetz, the Republican chairman of the Oversight Committee.
China urges North Korea to not violate United Nations resolutions again
South Korean activists regularly send propaganda balloons into North Korea, carrying leaflets or videos. Relations between Beijing and Seoul, strained by South Korea's hosting of the U.S.
Flynn was sacked in February for statements he made to Russian diplomats prior to Trump taking office.
That comment referred to revelations that he divulged classified information about an Islamic State threat in his May 10 meeting in the Oval Office with Russia's foreign minister and ambassador.
Mr Flynn's lawyers also said earlier in the day that the Justice Department's appointment of a special counsel has created a legally risky environment for him to co-operate with the Senate panel's investigation. However, it has not been alleged that Trump told the Russians that Israel was the source.
An additional complication for the congressional committees: If they move to grant Flynn immunity, they would probably have to enter into discussions with the special counsel, Robert Mueller, to determine whether that could impede the FBI's case.
Members of key congressional committees are pledging a full public airing as to why former FBI Director James Comey was ousted amid an intensifying investigation into Russia's interference with the U.S. election.
Comey was sacked by Trump earlier this month.
Former CIA Director John Brennan is to testify in open and closed hearings Tuesday before the House intelligence committee, which is conducting its own investigation.