University President Mary Sue Coleman announced the decision Tuesday but the choice of Brandon must be approved by the school's board of regents, which next meets Jan. 21.
If he passes the regents' vote, Brandon would begin his duties March 8.
Brandon replaces Martin, who announced his retirement as athletic director in October. He held the position since August 2000.
Brandon, 57, graduated from Michigan in 1974. He played football at the school under Bo Schembechler. He is currently chairman and chief executive officer of Domino's Pizza, which has its headquarters in Ann Arbor, Mich.
"It is my distinct honor and privilege to have this opportunity to serve the university in yet another way," Brandon said. "My participation as a student-athlete at (Michigan) has made a profound impact on my life and career and I fully understand and respect the important role our athletic programs play in helping to shape the culture and image of our university community."