Hamilton previously worked for Harbaugh at Stanford in 2010 as wide receivers coach, and the Wolverines have an opening for a quarterbacks coach and passing game coordinator.
The Browns don't have an offensive coordinator in title on their staff because head coach Hue Jackson handles many of those duties.
The 42-year-old Hamilton is Cleveland's highest-ranking offensive assistant but didn't call the plays, coaching the quarterbacks and helping coordinate the passing game.
Jackson could be replacing his top two assistants after the Browns' disastrous 1-15 season -- worst in franchise history.
Defensive coordinator Ray Horton is expected to be fired soon and possibly replaced by Los Angeles Rams DC Gregg Williams, a source told Cleveland.com. Williams reportedly has until Monday to decide if he wants the job.
Horton left the Tennessee Titans to join the Browns when Jackson was hired as head coach a year ago.