Kim had five birdies and a bogey in her 66 to join Grant on nine-under 131.
"Very, very solid," Grant said of her round on the par-70 Pelican Club layout in Belleair, Florida.
"I hit some really good shots. Tee to green today was very good. I felt like I played similarly to yesterday, but I just couldn't get the putts to go in. Today, a lot of them went in, so it was really fun out there."
Grant, a six-time winner on the Ladies European Tour, is chasing a second LPGA title to go with her 2023 Dana Open crown.
She said her miscue at her opening hole, the par-four 10th, actually helped her settle in.
"Sometimes, bogey on the first kind of helps you relax a little bit, just to get more motivated," she said. "And I think that was what it was today. I didn't hold on too tight in my putter. I just felt more relaxed."
It showed as Grant promptly birdied 11, 12 and 13. She added two more birdies at 16 and 18 and another three coming in.
"I felt like I scored kind of how I played," she said. "Still left a couple of putts -- you always leave a couple of putts out there."
Kim had five birdies in her first 11 holes, including three in a row from the ninth through to the 11th, before her lone bogey at the par-five 14th.
"It is nice to be able to kind of put some close and make the birdie putts," said Kim, who claimed her first major title at the Evian Championship this year. "I think that definitely helped."
The leading duo were one stroke in front of overnight leader Ryu Hae-ran, who carded a 68, and Jennifer Kupcho, who shot 66.
Ryu had four birdies and two bogeys in her two-under effort. She delivered another crisp performance off the tee, but unlike on Thursday, when she signed for a 64, she was unable to capitalise on several birdie chances.
"I think my score is not bad for afternoon tee time, but today's shots were so good (and) I missed a lot of birdie chances," the South Korean said.
"The greens were pretty good today, but afternoon tee time has to be a little bit more bumpy greens, so that's why I missed putts a lot today."