Bryant, the 2007 U.S. Senior Open champion, capped his round with a chip-in birdie on the par-5 ninth to grab a share of the lead. Kite missed a par-saving putt on the final hole to drop back into the tie.
Both finished 36 holes at 8-under 136. After Saturday's play at the Del Monte course, Sunday's final round will be played at Pebble Beach.
Sharing third place at 5-under 139 were Bobby Clampett (4-under 68), Mark McNulty (71) and Peter Senior (70).
First-round co-leader John Cook withdrew due to a neck injury.
2014: The Year in Music [PHOTOS]