American Jessica Korda hit a third-round 67 to move to third place on nine under.
“It’s a major so you know the girls up front are going to be nervous, and we’re kind of chasing them down and they know that,” said Korda.
“You always just kind of got to think that you have a chance no matter, and that’s kind of the mentality you got to go in there with.”
Overnight leader Hinako Shibuno, of Japan, dropped down the leaderboard after a 77 put her on four under.
England’s Georgia Hall also fell down the placings to three under after a 74 and she was leapfrogged by compatriot Charley Hull, whose 69 moved her to five under.
Northern Ireland’s Stephanie Meadow produced five birdies on the way to a 67 and is four under.
Defending champion Patty Tavatanakit (70) is in second place on 10 under.
“It was a really fun round out there. It was nice to see the putts start falling,” said 24-year-old Kupcho.
Eight of Kupcho’s birdies came on the first 12 holes and, despite a bogey on the 13th, helped put her in a strong position to land her first LPGA Tour title.
“I didn’t really look at scoreboards out there,” she added. “The first time I looked was on 13 to see who was in second, and it was Patty.
“Other than that, I’m just out there playing my game.metasports