Sen. Richard Burr, the chairman of the Senate intelligence committee, said the panel was reviewing a range of options to compel Flynn to disclose records about his meetings with Russian officials, including holding Flynn in contempt. And he said the panel "could" call for Flynn to assert his right against self-incrimination in a public session.
"It does us no good in having people pleading the Fifth if we are trying to get information," Burr said. He added: "The only thing I can tell you is immunity is off the table."
Unlike Flynn, two other former Trump campaign officials have turned over documents to the committee related to its investigation of Russian meddling in the US election.
"The subpoena seeks to compel (Flynn) to offer testimony through the act of producing documents that may or may not exist. In these circumstances, (Flynn) is entitled to, and does, invoke his Fifth Amendment privilege against production of documents," Flynn's lawyer wrote in a letter to the committee, verified by CNN.
When asked by CNN whether his panel would hold Flynn in contempt, Ranking Democrat, Sen. Mark Warner, said, "We have to find out whether we have the ability to either hold Gen. Flynn in contempt or whether it's just Fifth Amendment. I've got to get the legal answer to that first."
Warner said he was "disappointed" Flynn didn't produce the documents that were requested, and the committee was still determining if it had other options to get a hold of them.
Flynn's refusal to cooperate comes as he faces scrutiny in several inquiries, including by Capitol Hill and a federal grand jury that has issued subpoenas to associates of the ex-national security adviser.
His refusal will also intensify scrutiny over Trump's decision to hire him initially for the job and his decision to keep him on staff for 18 days after the President was warned by former acting Attorney General Sally Yates that Flynn may have been compromised by the Russians. And the White House is also bound to face new pressure from Trump's apparent disclosure of classified information in a private meeting with Russian officials at the White House earlier this month.
Warner, in an interview with CNN on Monday, said the White House must turn over records related to that meeting -- and did not rule out issuing a subpoena for the records.
"We have made the request," Warner said. "Transcripts or tapes. My hope is the White House will comply. They say there is no there there. They ought to work with the committee."
Former Trump campaign chairman Paul Manafort has turned over documents to the committee, a source familiar with the filing told CNN on Monday. Former Trump campaign adviser Roger Stone has also complied with the committee's requests and answered its questions, according to Stone's attorney, Robert Buschel.
The Senate committee had asked Flynn earlier this month to produce all records over his communications with Russian officials by this Wednesday. But Flynn is expected to send a letter later Monday invoking his Fifth Amendment rights.
The source close to Flynn said it would be "highly imprudent for him not to exercise his Fifth Amendment rights" given that several members of Congress have called for his prosecution.
Flynn's decision to decline the subpoena does not come as a surprise to Senate intelligence leaders, as Flynn's lawyer also told the panel last month he would not provide documents in response to an April request.
Interactive: The many paths from Trump to Russia
Flynn was back in the news last week following the revelation that former FBI Director James Comey wrote in a memo that Trump had asked Comey in a meeting to end his investigation
into the former national security adviser.
Comey has agreed to testify before the Senate intelligence panel after Memorial Day
. Warner said he and Burr hoped to meet with Robert Mueller, the Justice Department's special counsel in the Russia investigation, to talk about what Comey can and cannot say in his testimony.
"Our hope is that the chairman and I will be able to sit down and again just kind of get the rules of the road for us going forward with Director Mueller," Warner said.
Flynn resigned from the Trump White House in February
after it was revealed he'd misled White House officials over his conversations with Russian Ambassador Sergey Kislyak, which included communication about sanctions.
In what could be a signal of additional problems for Flynn, Democrats on the House Oversight Committee released a new letter on Monday calling for Chaffetz to issue subpoenas for White House documents.
The letter from Maryland Democratic Rep. Elijah Cummings
cites a Report of Investigation from the Pentagon, which was conducted by investigators undertaking a review for Flynn's security clearance application.
According to that March 14, 2016 report, Flynn made false statements to investigators
about who funded his foreign trips, including a 2015 trip to Russia where Flynn was paid roughly $45,000 for a speaking engagement. He claimed they were funded by "US companies," even though he was paid by Russia Today, according to the letter released Monday by House Democrats on the Oversight Committee.
The report stated that Flynn said he "had not received any benefit from a foreign country."
Flynn also claimed to investigators he had no substantial contacts with foreign government officials, saying he only had "insubstantial contact," but photos from the event showed Flynn sitting near Putin at that 2015 RT dinner.
The committee Democrats say Chaffetz must subpoena documents from the White House to determine what the White House was told about these foreign contacts and trips.
In the letter, Cummings demanded that Chaffetz either "issue a subpoena to the White House for the documents it is withholding or schedule a business meeting during which committee members can vote to issue the subpoena ourselves."
Flynn previously sought immunity
from the Senate committee in exchange for his testimony. Leaders of both the Senate and House panels, which are conducting separate investigations into Russia's election-year meddling, rejected that request.
During the 2016 campaign, Trump blasted aides to Hillary Clinton for taking the Fifth Amendment in relation to the investigation of her use of a private email server while secretary of state. He said at a September Iowa rally: "So there are five people taking the Fifth Amendment, like you see on the mob, right? You see the mob takes the Fifth. If you're innocent, why are you taking the Fifth Amendment?"
Sen. James Lankford, an Oklahoma Republican on the intelligence panel, said Flynn's decision would not stop the committee's investigation, tweeting: "It is Mike Flynn's right to plead the 5th. We will get to the truth one way or another. We need facts, not speculation & anonymous sources."
Flynn is one of several former Trump aides to whom Senate investigators have sent requests for information to as part of the panel's investigation into connections between Trump associates and Russian officials.
The panel has also sought documents from former campaign foreign policy adviser Carter Page.
The House intelligence panel, meanwhile, is requesting documents
from former Trump campaign communications adviser Michael Caputo.