Kim, the defending champion, shot a 2-under 68 to join Woods at 10-under 200. The two matched the tournament's 54-hole scoring record, which Tom Pernice Jr. set last year.
Michael Allen, who won the Senior PGA Championship in June, turned in the best round of the day -- a 5-under 65 -- to move into a share of third place at 211 with Cameron Beckman (66).
Woods will be seeking his 68th PGA Tour victory Sunday at Congressional. Kim will after his third tour title and first since last year in this tournament.