Norvell is expected to serve as receivers coach and passing-game coordinator for the Sun Devils.
He spent one season at Texas and served as interim playcaller for the Longhorns' final 11 games following the demotion of co-offensive coordinators Shawn Watson and Joe Wickline, whose contracts were not renewed after the season.
Coach Charlie Strong hired Tulsa's Sterlin Gilbert in December to take over the Texas offense.
Norvell's decision came on the same night that Texas parted ways with defensive backs coach Chris Vaughn. His contract was not renewed because of unspecified "circumstances," though his departure is reportedly tied to his alleged involvement in the NCAA recruiting violations levied against Ole Miss, where he coached from 2008 to 2011.
Texas lost running backs coach Tommie Robinson in January when he elected to rejoin the coaching staff at USC.
Following Vaughn's exit, only two coaches -- defensive coordinator Vance Bedford and linebackers coach Brian Jean-Mary -- remain from Strong's initial 2014 coaching staff.
WR coach Jay Norvell leaves Texas for ASU
ESPN Big 12 reporter Max Olson explains why Jay Norvell is the latest assistant to leave Charlie Strong's staff at Texas.