Rodriguez turned down an offer from South Carolina on Saturday to remain in the desert, Arizona athletic director Greg Byrne announced on Twitter.
MORE: Fired, retired, resigned coaches in 2015 | Meet the new FBS coaches for 2016
Rodriguez, 52, has a 32-20 record in four seasons at Arizona including a 6-6 mark this year. He led the Wildcats to a Pac-12 South title in 2014.
Before Arizona, Rodriguez had FBS stints at Michigan and West Virginia.
Auburn defensive coordinator Will Muschamp, Oklahoma offensive coordinator Lincoln Riley, former Rutgers coach Greg Schiano and California coach Sonny Dykes have also been linked to the South Carolina opening.