Spieth, 20, carded a four-under-par 69, as did defending champion Dustin Johnson, to sit 14 under overall alongside Simpson, who had four birdies on the back nine of his 68.
Leader Zach Jonson, meanwhile, could not consolidate the good work he had produced over the previous two days as two bogeys after the turn saw him manage just a 74 to slip to fourth at 12 under.
Dustin Johnson was heading nowhere at one over until an eagle at the ninth turned his form around, and he picked up a further three shots on the back nine.
Spieth and Simpson, meanwhile, each a carded a single bogey against five and six birdies respectively.
Behind Zach Johnson was a group of four on 11 under headed by Jason Dufner and Brandt Snedeker, while Masters champion Adam Scott was the leading non-American at 10 under.
Scotland's Martin Laird slipped a couple of places to 21st despite a 70, while Brian Gay had the round of the day with an eight-under 65.
At 12 over, Derek Ernst was the only man of the 30-strong field to be over par for the tournament.