Kim aced the 16th hole at the TPC San Antonio on Friday, and he missed it by inches on the way to a third-round 3-under 69 after opening with consecutive rounds of 66.
South Korean Kim - who became the youngest victor of the Players Championship in 2017 - carded a three-under-par 69 on Saturday.
"The first days here", Kim said, "I've been feeling good. I saw the bounce, but I never saw it go in".
Charley Hoffman made a Saturday surge with his back nine with a birdie-birdie-eagle finish to shoot the best round of the week at 8-under 64. He was two back of Kim at 13 under.
Joining Fowler and Spieth - both shot 68 - are Harold Varner (66), Adam Schenk (66), Kyoung-Hoon Lee (67) and Monday-qualifier Corey Connors (67).
Four strokes back from the leader at the start of play, Spieth went six-over 42 on the front nine, which included two double-bogeys and two bogeys without a birdie.
Kim's ace at the 167-yard 16th bounced up toward the flagstick, landed next to the lip of the cup and dropped in.
The eagle came after Fowler's drive had carried 328 yards to within four feet of the hole. He missed, Conners made his from inches shorter and they shared the lead again.
San Antonio - South Korea's Kim Si-woo will take a slender one-shot advantage into the final round of the Valero Texas Open after a birdie on the 18th hole left him within sight of a wire-to-wire victory on Saturday. "That was a heck of a shot".
Kim, victor of The Players Championship in 2017, got out of a greenside bunker at the par-5 18th and sank a four-foot birdie to get the one-stroke advantage. "I was just expecting the wind to hold it a bit better, and obviously it didn't".
Hoffman hit his approach shot with a 3-wood from 287 yards to within 10 feet on No. 18. "That's why it was a pretty good back nine".
Graeme McDowell, who won for the first time since 2015 last week at the PGA Tour's event in the Dominican Republic, has shot consecutive 69s and is joined at 6 under by Scott Brown, Hank Lebioda, Ryan Moore and Jhonattan Vegas. Before that, no one had done it in 24 years.
The 2017 Players champion spurned a glorious chance to birdie the par-three 16th after he was a matter of inches away from making a hole-in-one there for the second day running, although he did birdie the last after a fine greenside bunker shot to stay on track for a wire-to-wire victory.
South Korea launches world’s first 5G phone
The service is also available in other major cities outside Seoul. "I'll be able to enjoy it in better resolution and speed". The Galaxy S10 5G puts the unrivaled power, speed and connectivity of 5G networks directly into the hands of consumers.