Kapur, who recently finished 23rd at the US Open, followed up his first round 73 with a two-under 69 in second round to get to even par and tied 24th place after 36 holes.
Meanwhile, Jeev Milkha Singh (74-72) and Gaganjeet Bhullar (75-73) continued their struggle and missed the cut.
Kapur, who is hoping to land a late place in the British Open with a top-4 not otherwise exempt and also finish in the top-10 overall, had four birdies on the third, seventh, eighth and 10th holes and looked set to get close to top-10 to make a charge over the weekend. But bogeys on 16th and 18th messed up his plans.
The burly American moved to ten-under par and extended his lead to three shots over leading home hope Victor Riu, but his advantage would have been considerably greater were it not for a seven on the par-five ninth hole.
Riu delighted the large galleries at Le Golf National, in Versailles, with a round of 67 which featured five birdies, whilst Marcel Siem is two shots back on five-under par after the German battled his way to a round of 72.