The 6-5, 224-pounder lost the starting job to Patrick Mahomes late last season.
Webb, who threw for 5,557 yards and 46 touchdowns in three seasons, will graduate in May. Earning his degree this spring lets him be a graduate transfer and be eligible immediately at his new school.
He announced his departure Wednesday, a day after the Red Raiders ended a 7-6 season with a 56-27 loss to No. 22 LSU in the Texas Bowl.
Webb started 14 games, including the 2013 Holiday Bowl in which he was named MVP in a 37-23 win over Arizona State.
Mahomes is this season's Football Bowl Subdivision leader in total offense.