Nunes said Tuesday that he expected the NSA to turn over the information about such "unmasked" names to the committee by Wednesday or Thursday of this week.
The embattled House committee is conducting one of three probes into the election campaign, its aftermath and potential contacts between Trump officials and Russians.
Congress is investigating Trump's unverified claim earlier this month that Trump Tower was wiretapped and has a larger probe into whether Russian Federation interfered in the presidential election and may have colluded with members of the Trump campaign.
Nunes told reporters Tuesday that he meant to reschedule Yates, Clapper and Brennan for an open hearing "as soon as we can get the questions answered from the Federal Bureau of Investigation director".
Nunes acknowledged Monday that he reviewed intelligence reports at the White House complex and met a secret source behind his statement that communications involving associates of President Donald Trump were caught up in "incidental" surveillance.
Swalwell suggested that Nunes was working on behalf of the Trump administration to minimize the findings of the Intelligence Committee's probe. The next day, Nunes went to the Eisenhower Executive Office Building next to the White House to meet with an unnamed source who showed him documents related to US intelligence surveillance. Finally, Spicer told ABC's White House correspondent Cecilia Vega that Yates's attorney had clarified the decision in a letter "in plain English".
Critics say his disclosure was an effort to justify Trump's accusations that former president Barack Obama had tapped Trump Tower during the campaign in 2016.
That goes for "Republican, Democrat, Obama appointee" and career civil servants, he added.
White House spokesman Hope Hicks confirmed the meetings to The New York Times.
The White House has defended Nunes' actions, saying he had done his job to investigate allegations of surveillance and was being up front with journalists about his activities.
"We believe that the department's position in this regard is overbroad, incorrect, and inconsistent with the department's historical approach to the congressional testimony of current and former officials", O'Neil wrote.
If Nunes was not acting at the instruction of the president or a top White House aide - a reasonable suspicion at this point - when he made public statements, based on the reports, that served Trump's interests, it's up to the congressman to prove otherwise.
White House press secretary Sean Spicer dismissed accusations that the White House was being evasive on the specifics of Nunes' visit but refused to provide details on who signed the California Republican into White House grounds, as would be required protocol. "We have no problem with her testifying, plain and simple".
O'Neil declined to comment Tuesday, and a Justice Department spokeswoman did not return a message seeking comment.
Yates was sacked in January as acting attorney general after she refused to defend the Trump administration's first travel ban.
Flynn was forced to resign after it was revealed he discussed lifting sanctions with Russian Federation while President Obama was still in office. Tuesday, the lawmaker said there is not need to recuse himself from the investigative proceedings.
"It would be safe to say we have had conversations with a lot of people, and it would be safe to say Gen Flynn is a part of that list", Burr said.