Ko, for whom gold in Paris would complete her Olympic set, maintained her fluent form this week as she made six birdies against two bogeys to reach nine under par at the end of the penultimate round.
Metraux, who surged into the lead on Thursday on the back of an Olympic nine-hole record of 28 in her second round, had mixed fortunes on Friday.
At one point the Swiss slipped to two-over-par for the day after a fourth bogey on the 13th, before bouncing back with a birdie on 14th and a thrilling eagle on the last hole to end a one-under-par round that put her on nine under.
"I stayed patient. I think that was the key for me today," Metraux told reporters.
"I know this course is very, very difficult and any hole can get to you at any point."
The Swiss kept her composure despite playing with a rain glove after forgetting her regular golf glove in the locker room, she said.
Metraux is an Olympic debutant after choosing not to go to Tokyo while chasing a spot on the prestigious LPGA tour, which opened the way for her sister Kim to represent Switzerland three years ago.
In contrast, Ko has been in the mix at each Olympic golf competition since the sport returned to the Games in Rio.
American Rose Zhang and Japan's Miyu Yamashita were tied in third at seven under par, with Zhang making an eagle on the 18th to stay in the medal hunt.
Close behind were Thailand's Atthaya Thitikul at six under and Mariajo Uribe at five under, with the Colombian aiming to end her professional career on a high in Paris.
Defending champion and world number one Nelly Korda had another inconsistent round but also remained within touching distance at four under in the tied-seventh.
France's Celine Boutier, who delighted the home crowd by cruising into a first-round lead on Wednesday, also clung on to the leading pack, with three birdies on the last three holes putting her four under alongside Korda and Chinese pair Yin Ruoning and Lin Xiyu.
Australia's Hannah Green, meanwhile, struck the best score of the third round, a six-under-par 66, to revive her medal hopes as she climbed to tied-11th at three under.