Democratic lawmakers in Ohio called for an investigation into the hiring process at the state Department of Transportation after a politician said in a Facebook post that young Republicans would get preference for summer jobs at the agency, raising questions about cronyism in its hiring.

A Republican member of the Marietta City Council, Cindy Oxender, wrote a Facebook post on March 29 that advertised six summer jobs for high school and college students with the Ohio department. Young Republicans who applied would have a leg up, she said.

“It’s good money, good experience, and a very nice résumé builder,” she wrote, adding between parentheses, “Preference is given to Republican youth on this!”

Ms. Oxender deleted the post after criticism from news media in Ohio and a request from the city’s legal director, Paul Bertram. But it brought unwelcome attention to the city where Gov. John Kasich, the presidential candidate, delivered his State of the State address on Wednesday night.