McDowell made two birdies and a lone bogey on the tough Albatross course of Le Golf National to sit at 5-under 208 while Sterne was less consistent, off-setting two bogeys and a double bogey with four birdies for a 71.
Bernd Wiesberger of Austria (68) shared third place with David Howell of England (69) and Richard Green of Australia (70), one stroke back.
Soren Kjeldsen of Denmark was tied with Simon Dyson of England for sixth at 3 under.
Francesco Molinari of Italy had the best round of the day with a 67 to climb into 16th place, four shots back, and give himself an outside chance on Sunday.
Overnight leader Fabrizio Zanotti of Paraguay had a 78 to slip to 27th, while Thomas Bjorn of Denmark held a two-shot lead after a birdie on the par-3 No 8 - where he nearly had a hole-in-one but the ball lipped out - before dropping four shots in the last four holes to fall to eighth place.
Martin Kaymer of Germany carded a 69 to sit tied for 19th with Ian Poulter of England and Miguel Angel Jimenez of Spain. The 2010 US PGA Champion still hopes to secure a top 10 finish despite feeling "very sick."
"The first day I was OK, but yesterday I felt terrible. And today I just feel very weak," Kaymer said. "I don't hit the ball as far as usual."
Matt Kuchar of the United States virtually fell out of contention with a 73 that left him 10 strokes off the pace.
"Below-average golf for me," Kuchar said. "I haven't been as sharp as I had hoped to be, but I still enjoyed my stay here."
Ranked fifth in the world, Kuchar is making his French Open debut. He brought his family with him to France and will enjoy a short vacation next week in the south of the country.
He said his goal for Sunday is to "just have a good round, try to get on some better form. Then, hopefully carry some momentum into the British Open."
Leading scores after the third round in the French Open at Le Golf National outside Versailles on Saturday (GBR & IRL unless stated, par 71):
208 - Graeme McDowell 69-69-70, Richard Sterne (RSA) 68-69-71
209 - Bernd Wiesberger (AUT) 70-71-68, David Howell 69-71-69, Richard Green (AUS) 69-70-70
210 - Simon Dyson 70-68-72, Soren Kjeldsen (DEN) 69-68-73
211 - Thomas Bjorn (DEN) 68-69-74, Jorge Campillo (ESP) 74-69-68, Hennie Otto (RSA) 71-71-69, Lee Slattery 71-70-70, Marc Warren 69-72-70, Jamie Donaldson 70-70-71, Eduardo De La Riva (ESP) 72-67-72, Graeme Storm 70-68-73
212 - Francesco Molinari (ITA) 71-74-67, Simon Khan 71-73-68, Gregory Bourdy (FRA) 69-73-70
213 - Ignacio Garrido (ESP) 75-70-68, Miguel Angel Jimenez (ESP) 69-76-68, Ian Poulter 73-71-69, David Drysdale 70-74-69, Martin Kaymer (GER) 68-76-69, Chris Doak 71-70-72, Richie Ramsey 69-71-73, Stephen Gallacher 68-70-75
214 - John Parry 74-70-70, Ross Fisher 70-73-71, Rafa Cabrera-Bello (ESP) 70-72-72, Kristoffer Broberg (SWE) 72-69-73, Felipe Aguilar (CHI) 68-72-74, Fabrizio Zanotti (PAR) 68-68-78
215 - Luke Donald 71-73-71, Gareth Maybin 71-73-71, Robert-Jan Derksen (NED) 72-70-73, Scott Henry 73-68-74, Joost Luiten (NED) 71-71-73, Matteo Manassero (ITA) 73-69-73, Gareth Shaw 72-69-74, Matthew Nixon 68-73-74