Dallas County records show Kleysteuber, of Weatherford, was still being held in jail. Dallas police Cpl. Janice Crowther confirmed Tuesday that a person accused of stalking Hutchison had been arrested but would not provide any further details.
It was not immediately clear whether Kleysteuber had an attorney. No telephone number for Kleysteuber could be found.
There is no indication Kleysteuber came in direct contact with Hutchison. But he is accused of threatening her husband, Ray, at their Dallas home two weeks before he was arrested, according to an arrest warrant obtained by the newspaper.
In a statement, Hutchison's office said they reported a "potential concern" to U.S. Capitol Police before Congress began its two-week Easter recess last month.
Aides to the Republican said the danger was significant enough to warrant extra protection while Hutchison was in Texas.
"I am grateful to the Capitol Police and the Dallas Police Department for their skill and dedication and thankful that my family is safe," Hutchison said in a statement.
Hutchison has authored bills aimed at toughening stalking laws. She has said she was stalked in the 1970s by a man who broke into her campaign office and stabbed a poster of her with an ice pick.