Vovan and Lexus say the affair was set up in 2013 during a trip to Moscow and was arranged through Ksenia Sobchak, Russian President Vladimir Putin's goddaughter, who passed the photos to Putin.
In the audio, Schiff repeatedly asks for specifics about the nature of the meetings.
"I'll be in touch with the FBI about this. And we'll make arrangements with your staff. I think it probably would be best to provide these materials both to our committee and to the FBI," Schiff says in the audio.
The recording shows the duo saying they had audio of a conversation between a Russian singer who is a spy and former Trump campaign adviser and former National Security Adviser Michael Flynn.
Flynn resigned less than a month into the Trump administration after failing to fully disclose the nature of phone calls to Russian Ambassador Sergey Kislyak.
Late last year, Flynn pleaded guilty to lying to the FBI and is now cooperating with special counsel Robert Mueller's investigation into Russian interference with the 2016 election.
But a spokesperson for Schiff said the California Democrat had a feeling the call was bogus.
"Obviously, it was bogus — which became even more evident during the call — but as with any investigation that is global in scale, we have to chase any number of leads, many of which turn out to be duds," the spokesperson said.