UM Athletic Director Bill Martin made the announcement Sunday, saying Rodriguez, 44, will be introduced at a news conference Monday morning.
The Sunday Gazette-Mail in Charleston, W.Va., reported Rodriguez informed his players at West Virginia in a meeting Sunday afternoon, and the players filed out of the meeting room, somber-faced with little comment.
Rodriguez constructed a 60-26 record at Morgantown, including three straight 10-win seasons and two BCS bowl trips.
A report by the Detroit Free Press Friday said Rodriguez was in Toledo, Ohio, meeting with Martin and Michigan President Mary Sue Coleman to discuss the Wolverines' vacancy.
Rodriguez succeeds Lloyd Carr, who retired Nov. 19 in the wake of criticism following a fourth straight loss to arch rival Ohio State.
Nothing has been said about whether Rodriguez will coach the Mountaineers in their Jan. 2 contest against Oklahoma at the Fiesta Bowl.