Lee shares second with Englishman Paul Casey, with American duo Stewart Cink and Kevin Kisner a further shot back.
Also in contention, but needing an excellent final round, is rising Spanish star Jon Rahm at five-under, while Sergio Garcia and Jordan Spieth are at four-under.
They are all chasing Simpson, who broke away from the chasing pack with a three-under 67.
Simpson leads after three rounds for the fifth time in his career, and first since the Shriners Hospitals for Children Open in October 2013. That was also the last of his four PGA Tour victories. He lost in a playoff to Hideki Matsuyama at Phoenix in February.
"Definitely feel like (the season has) ebbed and flowed," he said. "I feel like that's starting to happen more, where I'm part of the conversation of contending. That's a good feeling. I know what I'm doing is working."
Lee tees off alongside Simpson in the final group at 5.15am (NZT).
- NZ Herald and AP