The Senate committee first requested documents from Mr Flynn in an April 28 letter, but he declined to cooperate with the request.
Rep. Adam Schiff of California, the ranking Democrat on the House intelligence committee, says The Washington Post's report that Trump tried to enlist the head of the National Security Agency and the national intelligence director to push the White House narrative is a "disturbing allegation" that Trump is interfering with the Federal Bureau of Investigation probe.
The letter also cited the unprecedented scale of the media campaign over the alleged Russian intervention, fueled by leaks from within the military-intelligence apparatus. "He is the target on almost a daily basis of outrageous allegations, often attributed to anonymous sources in Congress or elsewhere in the United State Government", the letter states.
Legal experts earlier predicted Mr Flynn was unlikely to turn over the documents without a grant of immunity because doing so might compel him to waive some of his constitutional protections.
Flynn was the first victim of Trump administration leaks after he told Vice President Mike Pence that he did not discuss sanctions with a Russian ambassador in a conversation before the election.
Flynn's contacts with the ambassador, according to officials who have previously described the communications, involved discussions about sanctions the Obama administration imposed on Russian Federation.
Flynn's attorneys, of the law firm of Covington and Burling, wrote that they believed it was reasonable to assume that any testimony or documents that Flynn were to provide could be used against him in a criminal proceeding.
That's according to a letter obtained by The Associated Press that was written on behalf of the former national security adviser under President Donald Trump.More news: Cher delivers homage to her best looks, songs
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While the attorneys had previously telegraphed Flynn's intentions, the letter delivered Monday officially set in motion a drama that's rooted in history but could lead to several potential climaxes.
Flynn received payments from Inovo BV, a Dutch consulting firm with ties to Turkish President Recep Erdogan, while he worked on the Trump presidential campaign in 2016. In fact, the oversight committee released detailed email and payment records months ago showing that the source of Flynn's payment of more than $33,000 was RT, the Russian state-sponsored television network that has been labeled a propaganda network by USA intelligence.
"If I were president-elect of the United States, I wouldn't let General Flynn in the White House", Christie said.
The committee, according to Flynn's lawyers, requested all records of his communications with President Donald Trump's campaign that were in any way related to Russian Federation from the day Trump announced his candidacy in June 2015 through Inauguration Day.
Representative Elijah Cummings said Donald Trump's former National Security Adviser told the investigators during an early 2016 security clearance review that a trip to Moscow was "funded by U.S. companies".
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.
Coats and Rogers declined the request, the Post reported. The requests could well be considered additional acts of obstruction of justice by Trump in the investigation by special counsel Robert Mueller.