The 22-year-old Sydneysider made eight birdies in a six-under 65 third round to surge to equal second at Highland Meadows on Saturday, on track for her best result on the tour.
Accomplished fellow Australian Minjee Lee was just a shot further back after a 67.
Kyriacou shares second spot with new US Women's Open champion Allisen Corpuz (68), Italy's Maria Fassi (67), Finland's Matilda Castren (67) and Denmark's Emily Pedersen (67).
But Grant goes into the final round with a six-shot lead at 18 under after just missing her chance to become the second LPGA Tour player to break 60 as she carded a 62.
Kyriacou had no doubt where her improved scoring has come from.
"I actually changed putters for the first time in like 12 years, so it was either going to be really good or really bad. So I'm glad it was really good," she said.
"I holed some nice putts, I've given myself some opportunities as well, which has been good but really converted them."
Nine under on her first 13 holes, 24-year-old Grant played the final five in even par. Fellow Swede Annika Sorenstam is the lone LPGA Tour player to shoot 59, accomplishing the feat in the 2001 Standard Register Ping at par-72 Moon Valley in Phoenix.
"There was a spectator who called it out in my face," Grant said when asked if she thought about a 59. "He just came up to me and he said, 'Do you think you have a 59 in you?' And then all of a sudden I was like, 'Oh, God.' So, I just tried to just not focus on it at all."
Seeking her first LPGA Tour victory, the 24-year-old Grant won the Jabra Ladies Open in May in France for her fifth Ladies European Tour title in two seasons.
Grant opened with a par Saturday and birdied the next four. After a rain delay of three hours, 48 minutes, she resumed on No.8 and added a birdie on No.9 for a front-nine 29.
Grant holed out a full shot from the fairway for eagle on the par-4 11th and birdied Nos. 12 and 13.
Needing to play the final five in three under for a 59, the former Arizona State player parred the next two, bogeyed the par-4 16th, birdied the par-5 17th and parred the par-5 18th.
with AAP
