For most of this year, Mr. Greitens, a former Navy SEAL who was elected governor in 2016, has been the target of investigations, charges that he is unfit to serve as governor and calls for his resignation or impeachment, even from members of his own party.

His troubles burst into public view in January when a St. Louis television station broadcast a report that before he became governor, Mr. Greitens had an extramarital affair with a woman, had taken a nude or seminude photograph of her without her permission, and had threatened to make the image public if she told anyone about him.

In February, the governor was indicted by a grand jury in St. Louis on a felony charge of invasion of privacy relating to the photograph. Separately, a House legislative committee in Missouri, mostly made up of Republicans, is investigating Mr. Greitens and could begin a process to impeach him at any time. Mr. Hawley, who has called for Mr. Greitens to resign, said on Tuesday that he had also referred evidence related to the Mission Continues donor list to the committee.

A report released by the House committee last week contained even more explosive allegations against Mr. Greitens. The woman who had a sexual relationship with him in 2015 told the committee that he hit her and she felt coerced into performing oral sex.

Mr. Greitens, a father of two, has insisted that the affair was consensual and that he and his wife, Sheena, have made peace with it.

Mr. Greitens faces trial on the invasion-of-privacy charge in St. Louis on May 14, and has said he views the hearing as an opportunity to clear his name. He could face up to four years in prison if convicted.

Missouri Republicans are concerned that the allegations swirling around the governor could hurt the party this fall in a competitive Senate race in which Mr. Hawley is hoping to unseat Senator Claire McCaskill, a Democrat.