Al Watkins represents the man who he said, "expended significant sums of money" to make sure the story didn't go public.
The recording is a conversation between the woman and her ex-husband, who secretly recorded the conversation as he pushed her to confess to the affair.
It is the Missouri governor's second marriage.
In it, she can be heard telling how Greitens invited her to his home in 2015.
"I would like to let the investigation play out and see what the real allegations are before I would do that", Beatty said. Once she arrived, he told her he would show her how to do pullups, taped her hands to exercise rings and blindfolded her, all with her consent.
She said she later realized he took a photo of her.
"If these allegations are true, proven true, I think he should resign".
She said no complaint was filed in her office, nor evidence of a crime presented.
The letter signed by both Democrats and Republicans called the allegations "deeply disturbing" and said a swift investigation was necessary to avoid having the matter "overpower" the 2018 legislative session. Greitens has denied any blackmail threat, reports said. On the Democratic side, the calls for resignation began last night with Democratic State Senator Jamilah Nasheed, who tweeted, "WE MUST END THE CULTURE OF CORRUPTION", a none-too-subtle reference to Greitens' frequent calls to rid Jefferson City of those aforementioned career politicians.
Also on Thursday, Greitens' attorney, Jim Bennett, released a statement denying any violence by the governor.
The woman, who has not been named publicly, was Greitens's hair stylist, according to media reports confirmed by The Washington Post with a source familiar with the situation. He said they also were looking to interview the ex-husband.
Additionally, she has stated that Greitens slapped her when she revealed that she had sex with her own husband. He did not say if the ex-husband has also heard from the Federal Bureau of Investigation.
Updated January 11 at 4:20 p.m. with Gardner investigation - Missouri House and Senate Republican leaders issued nearly identical statements of concern Thursday as they otherwise declined comment on the sex scandal swirling around Gov. Eric Greitens. We did not believe it was an ex-husband's story to tell. It is unclear whether the woman in question plans to come forward. His wife, Sheena Chestnut Greitens, is an assistant professor of political science at the University of Missouri, and a nonresident senior fellow at the Center for East Asian Policy Studies at the Brookings Institution in Washington. The couple has two sons, Joshua and Jacob. News 4 is not naming the woman and she has not made an on-the-record comment about the story.
Watkins said his client and the woman who had the affair tried to reconcile but that the relationship with Greitens continued for several months and caused their breakup.