The Briton has finished second two times in major tournaments, including the 2010 British Open. That year he also held the third-round lead at The Masters, only to finish runner-up to Phil Mickelson.
Mahan, also seeking his initial major victory, fired a 3-under 68, while three-time winner Woods turned in a round of 1-over 72 to end Round 3 at 1-under 212.
Reigning Masters champion Adam Scott claimed sole possession of fourth place at even-par 213.
Mickelson (72) and Francesco Molinari (72) were tied for ninth.
Notable deaths of 2014 [PHOTOS]
LeBron James reveals his new Nike shoes