American Schauffele, the world's form player after winning his maiden major titles at the PGA Championship and the British Open this year, carded a three-under-par 68 in his third round at Le Golf National.
He struggled to find his best during a level-par front nine as Rahm surged into the lead, but eagled the par-five 14th hole at the same time as Rahm was making a three-putt bogey on the next green.
Schauffele will likely have to endure another dramatic final day if he is to win gold again, having edged to victory in the Tokyo Olympics three years ago by a single stroke.
"I haven't gotten too far, thinking that far," Schauffele said when asked where he would put a second Olympic gold medal.
"But if I was to take a wild guess, I'm sure it would go right next to where the first one is."
Rahm started the day two shots off the pace but moved up the leaderboard with an excellent 66 to draw alongside Schauffele on 14-under for the tournament.
The two-time major champion made seven birdies, including a long putt on the penultimate hole to bounce back after his disappointment on the 15th.
Rahm is back at his best after a slow start to the year following his switch to the Saudi-backed LIV Golf circuit.
He posted his first top-10 finish at a major this season last month at the British Open, then sealed his first LIV victory in England last weekend.
"It's nice to see things going so well for the last month, and after having won, to be in this position again so quickly," Rahm said.
Tommy Fleetwood is just one shot behind the leading duo in third place on 13-under after a 69 which featured three birdies and one bogey.
The popular Briton, who has never won a major title despite several near-misses, was not on top form but did enough to ensure he has a strong chance of clinching gold.
Hideki Matsuyama, tied for the overnight lead with Schauffele and Fleetwood, battled hard for a level-par 71 to remain on 11-under.
The Japanese is tied with Nicolai Hojgaard after the Dane powered through the field with a spectacular nine-under 62 to equal the course record.
The 23-year-old now jointly holds the record with three other players, including his twin brother Rasmus.
McIlroy, Scheffler in touch
Rory McIlroy produced his best round of the week so far with a 66 to stay in touch with the leaders, sitting level with world number one Scottie Scheffler on 10-under overall.
The Northern Irishman was bogey-free, holing a crucial 20-foot par putt on the 17th.
"I think tomorrow, depending on what the leaders do, I'm going to have to probably shoot my lowest round of the week to have a chance at a medal," said McIlroy. "That's the goal."
Scheffler bogeyed the 17th and then was in trouble down the final hole, but got up and down from 90 yards for par to boost his hopes of winning a medal on his Olympic debut.
"I'd like to be leading. I feel like I haven't had my best stuff the last few days but I've done enough to hang in there and stay in the tournament," said the Masters champion.
South Korean youngster Tom Kim, who would earn an exemption from military service with a medal, finished strongly to be tied for sixth with McIlroy, Scheffler and Belgium's Thomas Detry.
World number four Ludvig Aberg of Sweden, Australia's Jason Day and Chilean Joaquin Niemann are all at nine-under, five shots off the lead.
