LANSING, Mich. — As voters in Michigan go to the polls on Tuesday, Gov. John Kasich of Ohio is getting some help from Mitt Romney, who recorded a get-out-the-vote call for Mr. Kasich’s campaign.

Mr. Romney spoke out forcefully last week against Donald J. Trump, and he also recorded a phone message for Senator Marco Rubio of Florida, who, like Mr. Kasich, is trying to cut into Mr. Trump’s support. Mr. Romney’s calls are going to Republican voters in Michigan and the three other states voting on Tuesday: Mississippi, Idaho and Hawaii.

Mr. Romney’s father, George Romney, was Michigan’s governor, and Mr. Kasich has put significant effort into the state, which borders Ohio. He held a rally at a brewery in Lansing on Tuesday, then planned to return to Ohio, where he will spend primary night.

The call recorded by Mr. Romney on Mr. Kasich’s behalf is virtually identical to the one he recorded for Mr. Rubio. It focuses on undermining Mr. Trump, rather than making a case for Mr. Kasich.