As Southern states reconsider Medicaid expansion, Gov. Laura Kelly of Kansas is pushing her own plan meant to appeal to conservatives. So far, success has been elusive.

As lawmakers in a nearby hearing room debated last month whether to support her legislation to expand Medicaid, Gov. Laura Kelly of Kansas dared the state’s Republican House speaker to hold a vote.

“If he thinks he can kill it, bring it,” Ms. Kelly, a soft-spoken moderate Democrat, said in an interview in her sprawling office suite at the State Capitol in Topeka.