On Monday, Flynn invoked his Fifth Amendment right under the US Constitution not to provide the panel emails and other documents detailing his relationship with Russian officials as per the committee's subpoena.
Burr did not rule out further steps, including holding Flynn in contempt, if he continued to refuse to comply.
"Everything is on the table", he told reporters at the Capitol.
They were also sending two new subpoenas specifically targeting Flynn's businesses, Flynn Intel LLC, and Flynn Intel Inc., both based in Virginia.
Citing three current and former American officials, the Times said United States intelligence officials collected information last summer showing Russians zeroed in on Trump campaign chairman Paul Manafort and Mr Flynn, a former head of USA military intelligence who was a key Trump campaign adviser. Burr said that the committee's lawyers have supplied Flynn with a narrower set of questions.
"He ought to comply with their request for documents".
In his refusal to turn over the documents, Flynn claimed his Fifth Amendment right against self-incrimination, and argued that an "escalating public frenzy" has emerged around the investigation.
"It is not our plan today to offer immunity, " Burr said. "I don't know. But I know that there was a sufficient basis of information and intelligence that required further investigation by the bureau to determine whether or not USA persons were actively conspiring, colluding with Russian officials", Brennan said.
In March, Sessions was harshly criticized for claiming during his Senate confirmation hearing that he had had no communication with Russian Federation despite those interactions with Kislyak. One remedy available may be a resolution to hold Flynn in civil contempt of Congress. That was the approach taken by the Permanent Subcommittee on Investigations of the Homeland Security and Governmental Affairs Committee in dealing with the website Backpage.com.
The committee sent a letter to Flynn's lawyer Tuesday addressing concerns that its original subpoena lacked specificity. John McCain, R-Ariz. The Supreme Court, in a one-page order, declined to stop the subpoena from being enforced.
Testifying before a House panel, former Central Intelligence Agency director John Brennan said he was aware of contacts between Russian officials and people involved in the Trump campaign and grew concerned that Moscow had sought to lure Americans down "a treasonous path".
"Frequently, individuals who go along a treasonous path do not even realize they're along that path until it gets to be a bit too late", Brennan said.
Sessions recused himself in March from an investigation into Russian interference in the presidential election following the revelation of two previously undisclosed encounters with the ambassador, Sergey Kislyak.
On Tuesday, former CIA Director John Brennan told lawmakers he had noticed contacts between associates of Trump's campaign and Russian Federation during the campaign and grew concerned Moscow had sought to lure Americans down "a treasonous path".
"I know what the Russians try to do", he told the committee.