He was at 15 under and Connors, trying to become the first Monday qualifier to win on the PGA Tour since 2010, was a stroke back after shooting a 6-under 66.
Kim, 23, held a narrow one-shot lead overnight following his first-round 66.
Kim, appeared to be struggling a little with a rib injury in the closing stages but he still had three birdies in four holes from the 14th after his earlier troubles before bogeying the 18th.
"I'm happy with it", Fowler said of the round. Just yesterday he switched his technique of reading greens after experiencing what he thought was shaky nerves over putts. "I'm not looking to be validated or not validated, I'm just looking for progress".
Spieth matched his first-round score of 68 with the help of an eagle and five birdies but he also bogeyed the first, 10th and 12th holes. "Gave myself a ton of chances, hit a lot of quality iron shots set up by some good tee balls". "That's why it was a pretty good back nine".
Behind the streaking Conners is a tour veteran who is in contention for his first win since claiming the Valero Texas Open championship in 2016.
He remained on track with a birdie at the 14th after getting up and down from a greenside bunker and, after holing a clutch 12-footer for par at the 15th, he notched another birdie at the par-three 16th where he almost took out the flag with his tee shot. He's two ahead of a group that includes Fowler and Jordan Spieth at the PGA Tour's last stop before Augusta.
"I've been in this position a few times before where I've been near the lead", Conners said.
"Flushed it", said the laid-back Californian.
Jones was bogey-free today and heads into the weekend well placed to improve on the consecutive top twenties he has recorded in his last two starts.
"So normally the better results come from the club starting to feel more in control".
The Valero Texas Open is played on the AT&T Oaks Course at TPC San Antonio, in Texas.
An finished his day tied for 17th at 3 under; Si Woo Kim is in 1st at 6 under; Abraham Ancer, J.T. Poston, Brian Stuard, and Jhonattan Vegas are tied for 2nd at 5 under; and Jordan Spieth, Chris Stroud, Roberto Díaz, Hank Lebioda, Max Homa, Ryan Moore, Scott Stallings, Danny Lee, Ted Potter, Jr., Morgan Hoffmann, and Rickie Fowler are tied for 6th at 4 under.