Ryu nearly finished with a 59 -- unbeknownst to her at the time -- but her 30-foot putt for an eagle on the par-five 18th hole on Saturday came up just short.
The South Korean finished with her fourth birdie over the final five holes at the Evian Resort Golf Club in Evian-les-Bains to move to 19-under 194 at the fourth of this season's five majors.
The winner of last month's Women's PGA Championship in Chaska, Ryu is three strokes clear of Japan's Akie Iwai (65 on Saturday) heading into Sunday's final round.
Ryu admitted that she did not know she was on the verge of a sub-60 round until she counted the scorecard after her final putt.
"I don't count my score on the green today, so that's why I don't know on the hole 18," Ryu said.
"But after the putt and I counted my score with my caddie. Oh, my God, it's 11-under par today. It was so amazing. My caddie says, 'yep'. Yeah, I'm so happy right now."
Ryu recorded nine birdies and an eagle on the par-four sixth hole en route to her 60, which was one stroke better than a trio of other finishes in major championship history -- all at the Evian.
Hyo Joo Kim set the record in 2014 before Jeongeun Lee6 and Leona Maguire matched it in 2021.
The 25-year-old Ryu now finds herself on the verge of capturing her second major in three weeks. The win at the Women's PGA on June 28 was the first victory in a major.
"Yeah, that is amazing, amazing dream," she said. "So I just want to that one come true, but we have one more day and Aki is pretty good player and everybody is so good."
Iwai collected eight birdies against two bogeys to remain in contention, while Canada's Brooke Henderson (64) and Japan's Mao Saigo (67 sit in a third-place tie, seven strokes behind Ryu.
South African Casandra Alexander jumped to fifth at 11-under by carding a 63, while Thailand's Jeeno Thitikul (67), Japan's Miyu Yamashita (67) and second-round leader Lottie Woad (72) of England sit in a tie for sixth place at 10-under.
Australia's two remaining weekend survivors, Cassie Porter and Karis Davidson, battled well on moving day.
Davidson shot a five-under 66 to move up to joint 16th at six-under, but she was still 13 shots off the pace, while Porter was a couple of shots further back at four-under after a 68.
