The suit comes after Michigan's elections board deadlocked on Trump's request to prevent a recount, which means it will start next week unless the courts intervene.
Two Republicans voted Friday to prevent the recount, while two Democrats said it should proceed. A state spokesman says the statewide recount requested by Green Party candidate Jill Stein will begin Tuesday or Wednesday barring a court order.
Trump's suit was followed a few hours later by a lawsuit from Stein. She is suing in federal court in an effort to get the two business day waiting period overturned.
Michigan Attorney General Bill Schuette, a Republican, asked the state Supreme Court on Friday to intervene and stop the recount. That motion is pending.
A Wisconsin recount is underway, though his supporters have filed a federal lawsuit trying to stop in. In Pennsylvania, Trump is asking a court to dismiss Stein's recount request.