Labor unions have rejected an offer by Ohio Gov. John Kasich to seek a compromise on a new law that removes most collective-bargaining rights for the state's 350,000 public employees, as a fight over the legislation heads toward a statewide referendum in November.

On Wednesday, Mr. Kasich, a Republican, and the party's leaders in the Ohio Senate and House made a pitch to public-employee union leaders to "avoid the bitter political warfare" over the law, known as Senate Bill...