He said they also were looking to interview the ex-husband.
Gardner added, "It is essential for residents of the City of St. Louis and our state to have confidence in their leaders".
The governor and his wife released a statement on Wednesday saying he had been "unfaithful in our marriage".
In a recording provided to a St. Louis news station by her ex-husband, the woman says Greitens threatened to release nude images of her if she ever spoke out.
The advertisement featuring Greitens had him and the fellow Republican governors mockingly thanking Democratic House Speaker Michael Madigan for improvements in their states.
The New York Times reported upon Greiten's admission, which included his assessment of the affair as a "personal mistake". "Eric took responsibility, and we dealt with this together honestly and privately". This personal matter has been addressed by the Governor and Mrs. Greitens privately years ago when it happened.
A local prosecutor on Thursday launched a formal investigation into allegations that Missouri Gov. Eric Greitens (R) blackmailed a woman with whom he was having an affair.
An attorney for the woman's ex-husband says he has been interviewed multiple times by the Federal Bureau of Investigation. "After further consideration, I have made a decision to launch a formal investigation into the alleged actions of Governor Greitens".
On the tape, she said that he invited her downstairs at his home because he wanted to show her "how to do a proper pull-up".
The woman alleges Greitens taped her hands to some rings and blindfolded her.
She said she later realized he took a photo of her. "I believe it's going to keep growing - I don't think it's going to go away over the weekend".
The woman did not immediately return a call left by The Associated Press at the salon where she worked. A bipartisan group of Missouri state senators also asked the state's attorney general to investigate the governor. Attorney General Josh Hawley's office did not have an immediate response.
"The seriousness of this allegation and the implications it will have on the integrity of our state government are deeply disturbing", he wrote.
The statement reads: The serious allegations against Missouri Governor Eric Greitens are very troubling. Democratic Sen. Jamilah Nasheed and Democratic Rep. Brandon Ellington both said Greitens should resign.
The St. Louis Post-Dispatch reported that the email was sent October 20, 2015, three weeks after Greitens filed papers formally starting his 2016 gubernatorial campaign.
The unnamed woman did not make the accusations of blackmail public. "We could either stick our head in the sand in the legislature or we can move forward".
Ex-Husband: For the first time? He also says it puts Missouri in a bad light nationally.
Bennett denied that authorities are involved.
"We have been asked repeatedly by reputable news outlets why we believe this almost three-year-old news story is coming out now".
Within 24 hours of acknowledging the affair in a contrite statement late Wednesday, Greitens' camp had shifted to suggesting he was the target of "a political hit piece" - even as the St. Louis circuit attorney launched a criminal investigation.