The American gathered five birdies in nine holes from the sixth en route to a 14-under 196 total with one round left at Firestone Country Club.
Northern Irishman Rory McIlroy also had a 67 to jump into second place with Poulter (70) at 11-under.
Tiger Woods plunged 11 shots behind after a poor 73.
Thomas and McIlroy are paired for the final round, potentially setting up a duel between the south Florida neighbours.
"The hard part about this course is you can get off track pretty quick if you're not playing well," Thomas told CBS television.
"I just need to continue to play well tomorrow because I have a lot of really good players chasing me."
Australian Jason Day (69) moved into fourth place, four shots behind. Nobody has come from further back in the final round to win the championship.
World number three Thomas has won eight times on the PGA Tour, but is still after his first World Golf Championships title.
He lost to Phil Mickelson in a playoff at the WGC-Mexico Championship in March.
McIlroy was satisfied with a bogey-free round, particularly the quality of his driving, as he smashed his tee shots more than 300 yards with monotonous regularity.
"I like to hit driver as much as possible and this course allows me to do that," he said.
"It allows you to be aggressive off the tee. (Hitting) nine out of 14 fairways today is a good number if I can do that tomorrow hopefully I'll have a chance."
As McIlroy crept up the leaderboard, Fleetwood regressed. He got an unlucky break at the first hole when his approach shot hit the flagstick and clattered off the green, which set the tone for a 74 that left him seven shots back.
Woods, an eight-times champion at Firestone, dismissed speculation that his surgically-repaired back had bothered him.
"I'm fine," he said. "I just played like crap."
Asked to specify, he said: "Everything. I didn't warm up very well, didn't hit the ball crisp or clean.
"Very similar to the first day. Difference is I made everything the first day. Today I didn't make anything."
(Reporting by Andrew Both in Cary, North Carolina, editing by Pritha Sarkar / Ian Ransom)
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