Walker, who lives 35 minutes away from TPC San Antonio, is looking for his fifth title in two tour seasons. He had a 9-under 207 total after opening with rounds of 71 and 67.
Spieth shot a 71. The 21-year-old Dallas player, coming off a playoff victory two weeks ago at Innisbrook, had a double bogey, two bogeys and two birdies on the final six holes.
FedEx Cup champion Billy Horschel birdied the final three holes for a 69 to get to 3 under.
