Kosar, 50, was charged in September in Solon, Ohio, with speeding and operating a vehicle while under the influence of alcohol. He refused to take a breath alcohol test when he was stopped, but police who pulled him over noted the smell of alcohol in his car, they said.
Kosar pleaded not guilty to both charges.
His lawyer appeared in court Monday and requested the rescheduling because Kosar is out of town, the (Cleveland) Plain Dealer reported Monday.
Kosar played for the Browns from 1985 to 1993.