Kush played center for California University, a Division II school in Pennsylvania.
The Chiefs lost center Rodney Hudson to a season-ending broken bone in his left leg three games into last season. Ryan Lilja slid over from guard to center for most of the season, but he announced his intention to retire after the Chiefs finished 2-14 a year ago.
The left the Chiefs in need of a backup center.
Kansas City spent its first two picks Saturday on defense: Alabama linebacker Nico Johnson was snapped up in the fourth round and Georgia defensive back Sanders Commings in the fifth.