Lawmakers have said that Mr. Greitens came off as smug at times and was quick to tout successes as his own. In a recent video “success story” on his Facebook page, he pours out a tin of car keys and says he will save taxpayers $500,000 by getting rid of 30 state-owned cars.

Most Republicans in the State Legislature have not pulled their support for the governor over the affair, but they have not loudly defended him, either, and some have already suggested he must resign — or, one said, be impeached — if all the allegations leveled against him prove true.

Mr. Greitens’s lawyer, James Bennett, said the governor was not resigning. He added that Mr. Greitens was paying him personally and that “his goal is to begin the process of making amends with people who may feel impacted by the lapse in his conduct that he has acknowledged.”

While Mr. Greitens has admitted to the affair, he denied that he had threatened to release a compromising photo of the woman if she told anyone, an allegation made by the woman’s former husband in a report that aired on KMOV, a St. Louis television station. The ex-husband leaked a recording of the woman in which she brought up the photo.

The woman, who has not been publicly identified and did not respond to requests for comment, requested privacy in a statement by her lawyer on Friday.

“It is very disappointing that her ex-husband betrayed her confidence by secretly, and without her knowledge, recording a private and deeply personal conversation and then subsequently released the recording to the media without her consent,” said the statement from her law firm, Knight & Simpson.

Some Democrats have called on Mr. Greitens to resign. Several Republican lawmakers signed a letter asking the state attorney general, Josh Hawley, to investigate the blackmail claims. Mr. Hawley’s office said the issue was outside his jurisdiction but St. Louis’s prosecutor, Kimberly Gardner, said she was investigating.