McDowell shot a 1-under 70 while Sterne had a 71 on the tough Albatross course of Le Golf National on Saturday, leaving both at 5-under 208.
"I couldn't have asked for a better position to be going into the last round," McDowell said. "The putter just has not quite got revved up to top gear yet, but hopefully that's waiting for tomorrow."
McDowell is rediscovering his form after missing the cut in his previous three tournaments.
"I bogeyed 17 probably off the best tee shot I hit all day, maybe just wrong club selection off the back there," McDowell said. "I tried to nudge the putter out of the fringe and it was a bit sticky and came out hot on me."
Francesco Molinari of Italy had the day's best round, a 67 that left him four shots back. Second-round leader
Thomas Bjorn of Denmark held a two-shot lead after a birdie on No. 8 -- where he nearly had a hole-in-one but the ball lipped out -- before dropping four shots in the final four holes.
Martin Kaymer of Germany, Ian Poulter of England and Miguel Angel Jimenez of Spain are part of a five-way tie at 213. Kaymer, the 2010 PGA champion, shot a 69 and is hoping for 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 fell from 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 take a short vacation next week in the south of the country.
He said his goal Sunday is to "just have a good round, try to get on some better form. Then, hopefully carry some momentum into the British Open."