The Senate Intelligence Committee subpoenaed Flynn's personal documents on May 10, after he declined to cooperate with an April 28 request from the panel in relation to its investigation of Russian interference in the election and possible ties to Trump associates.
Before the April request, Flynn said through a statement from his lawyer that he wouldn't submit to questioning from the committee "without assurances against unfair prosecution."
The committee leaders are directing the two new subpoenas at Flynn's Virginia-based businesses because corporations are not protected by the Fifth Amendment, Warner said.
“While we disagree with Gen. Flynn's lawyers' interpretation of taking the Fifth, it's even more clear that a business does not have the right to take a Fifth if it's a corporation. One subpoena has been served. One is in the process of being served," Warner said.
The committee sent a letter to Flynn's lawyer Tuesday addressing concerns that its original subpoena lacked specificity.
"We've been very specific in the documents now that we've requested from Gen. Flynn," Burr said.
A contempt charge is still a possibility.
"If in fact there is not a response, we will seek additional counsel advice on how to proceed forward. At the end of that option is a contempt charge, and I've said that everything is on the table," Burr said. "That is not our preference today. We would like to hear from Gen. Flynn. We'd like to see his documents. We'd like him to tell his story because he publicly said, 'I've got a story to tell.' We're allowing him that opportunity to do it."
But immunity is not on the table.
"It's a decision that the committee has made that we're not at the appropriate avenue in a potential criminal investigation. As valuable as Gen. Flynn might be to our counterintelligence investigation, we don't believe that it's our place today to offer him immunity from this committee," Burr added.
In testimony before the committee on Tuesday morning, former CIA Director John Brennan said he confronted a Russian intelligence chief about election meddling last summer, and Warner said the committee is looking into it.
"We have to make sure we don't see it coming forward again in the future. And what we're looking at now is to look at those contacts that Mr. Brennan spoke about and see what they were, how extensive they were and what they led to, if anything," Warner said.
ABC News' Matthew Mosk and Meghan Keneally contributed to this report.