Rahm gained on the field with a seven under 65 around PGA West's Stadium Course on Saturday, enough to tie up with Thompson on 23 under for the tournament, putting the duo four shots ahead of the chasing pack.
The indomitable Spaniard gained momentum on the front nine, rolling in four birdies and an eagle to make the turn at six under. He cooled down somewhat coming home, with two birdies and a lone bogey blemish on the 17th.
"Nothing but positive... ball-striking felt amazing," quipped the world number four.
"Hit a lot of great shots out there off the tee. Giving myself a lot of opportunities with the irons. Took advantage of quite a few of them. So very confident going into tomorrow and knowing that I'm going to have to shoot another low score if I want to have a chance to win."
For Thompson, his red-hot putter that's drained five eagles this week narrowly missed a 35-foot birdie opportunity on the last that would have given him a one-stroke lead going into Sunday.
"I feel like everybody would pick me as the underdog," he said of his chances at grabbing his first piece of PGA Tour silverware.
"But I mean, I kind of relish that label. I just try to go out and do my own thing and try to not let anything bother me."
Further down the leaderboard, South Africa's Christiaan Bezuidenhout is tied for third with JT Poston on 19 under, while the in-form Tom Kim lurks one shot back.
American Dylan Wu also deserves a mention, as he carded a sizzling round-of-the-day 11 under 61 to surge up the leaderboard to 17 under.