The University of Iowa says Nagel died Jan. 15 in San Antonio. His daughter Nancy Nagel confirmed her father's death when reached by phone Wednesday.
The Los Angeles native played quarterback for UCLA and began his coaching career there as an assistant.
He was head coach at Utah from 1958 to 1965 and at Iowa from 1966 to 1970.
He was inducted into the National Association of Collegiate Directors of Athletics Hall of Fame in 2006.