"I am extremely excited to join the University of Michigan and work with Brady Hoke, the staff and players," Nussmeier said in a release. "I'm proud of what we accomplished in two seasons at Alabama, and I owe a great deal to Coach [Nick] Saban for that opportunity.
"Michigan is a program I've always had deep respect for, and I'm looking forward to getting started in Ann Arbor and being a part of the great tradition there."
Borges' firing was announced Wednesday.
Nussmeier, 43, spent the previous two seasons as the offensive coordinator and quarterbacks coach at Alabama, working with Heisman finalist AJ McCarron and winning a national title in his first year with the Crimson Tide. He also was offensive coordinator at Washington (2009-11) and Fresno State (2008).
Nussmeier, who played at Idaho, is one of four college quarterbacks to pass for 10,000 yards and rush for 1,000.
"Doug is a highly respected offensive coordinator," Hoke said in a release. "Doug has been successful at every coaching stop with his balanced and explosive offenses, and he brings national championship experience. He is an excellent addition to our coaching staff and football program."