Bradley, seeking his second win at the Texas event and fourth title of the year, finished the third round with a one-stroke lead at 13-under-par 197.
Sang-Moon Bae grabbed sole possession of second place at minus-12 after firing a 4-under 66 at TPC Four Seasons Resort.
Tom Gillis, meanwhile, moved within two shots of the lead at 11-under 199 with a round of a 3-under 67.
Holding fourth place one shot further back was Scott Piercy, who carded a 66.
Sharing fifth at minus-8 were Charl Schwartzel (69), Gary Woodland (68), Harris English (68) and John Huh (69).
Swim Week Miami Beach 2014 [PHOTOS]