Projected starter Bourbon tore his anterior cruciate ligament and Cox tore his Achilles tendon, the team said.
Both players have already graduated with undergraduate degrees.
Junior college transfer De'Andre Mann and freshman Corey Avery now will move up the depth chart.
"I feel bad for both Brandon and Taylor as they were looking forward to the opportunity to be the replacement for James Sims," Kansas head coach Charlie Weis said. "I sat down with each of them individually and talked through their options and was happy that they chose to further their education, while attempting to play again next year if things work out."
Bourbon played in all 12 games last season, with six starts. He had 191 yards rushing with three touchdowns and caught 20 passes for 102 yards. Cox ran for 464 yards with three touchdowns and snagged three receptions for 24 yards.
Kansas opens the season at home against Southeast Missouri on Sept. 6.
- Scooby Axson