Svensson had four birdies in the first six holes and five overall. But he bogeyed the 17th at Le Golf National on the outskirts of Paris to finish the day 13 under overall.
Despite that bogey, the 28-year-old Svensson showed strong consistency by making 16 of the 18 greens in regulation.
"I hit the ball really well all day, a lot of good looks, hit a lot of good putts," he said. "A lot of putts slid past, hopefully they'll drop tomorrow."
Olesen had four birdies and also carded 67, while Bairstow's 6-under 65 included a superb chip-in eagle on the 14th hole.
"Front nine was very smooth and I just holed a few putts on the back nine," Bairstow said. "Hopefully keep it going."
England's Dan Bradbury, who shared the overnight lead with Svensson and Belgium's Thomas Detry, posted 69 and heads into Sunday's final round two strokes behind Svensson in a tie for fourth with countryman Joe Dean (66).
Australia's David Micheluzzi was well placed for some decent prizemoney after posting a second straight 67 to get to seven under and tied 25th.
Lev Grinberg, the 16-year-old Ukrainian playing for just the fifth time on the European tour, was undone by a triple bogey on the 15th - where he hit two balls into the water - and carded 70. He was tied for 30th.
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