The 62-year-old Kite won the 1983 Bing Crosby National Pro-Am and 1992 U.S. Open at Pebble Beach, the site of the final round Sunday in the Champions Tour event.
"I didn't play quite as well today tee to green as I did yesterday," said Kite, who won the last of his 10 Champions Tour titles in 2008. "But I made some nice recoveries. Beginning on the 11th hole, I began make some nice shots."
He's in position to become the third oldest tour winner behind Mike Fetchick and Gary Player.
"I'm pleased to be playing well," Kite said. "It's nice to do something and feel like I have a chance. Fifty-year-olds keep winning. It's more difficult, but I'm on a golf course I know well and like a lot."
Bryant, winless since the 2007 U.S. Senior Open, had a 67 at Del Monte to match Kite at 8 under.
The 58-year-old Bryant is playing with a ruptured tendon in his right foot.
"It hasn't affected my golf swing much," Bryant said. "But I was never in very good shape to begin with. ... Today was a very fortunate day. I didn't drive the ball well, but I managed the course well."
He has been plagued with his foot injury for several years and is considered curtailing his career.
Kite, tied for the first-round lead with John Cook after a 67 at Del Monte, had four birdies, but lipped out his par putt on the 18th for his only bogey of the round.
Cook withdrew after nine holes Saturday because of a lingering neck injury. He was 5 over for the round and even par overall when he stopped playing. Defending champion Jeff Sluman was 1 over after a 74 at Pebble Beach. He also won the event in 2008 and '09.
Eighty-one junior players from the First Tee program played alongside the pros.