Hamilton spent one season with the Browns, joining Hue Jackson's staff after he was hired as head coach.
On Sunday, Jackson said he was hopeful he would be able to keep Hamilton on his staff.
"We want him to stay. We hope he stays. Like I said, a young man is entitled to do what they feel is best for them, but I would think we have created a good environment, and I hope things are still good. As far as I know, things are," he said.
Hamilton will help fill a void left by former quarterbacks coach Jedd Fisch, who left the Wolverines to become UCLA's offensive coordinator last week.