Bourdy had an eagle on his second hole and six birdies and three bogeys to move to 12-under 132 after a 67. Englishman Mark Foster also shot a 67 and was in second.
Former U.S. Open champion McIlroy climbed 29 places and into a tie for 24th at 139 with his 3-under 69 with five birdies and two bogeys on the Royal Portrush links course.
Three-time major winner Padraig Harrington moved into contention with a second straight 67 to sit two shots off the lead and part of a three-way tie for third.
PGA champion Keegan Bradley missed the cut at his first tournament in Europe after a 1-under 71 on the second day to go with his 1-over 73 on Thursday for even par overall.
Bradley started with a double-bogey 7 on No. 10, his first hole, and despite a run of four straight birdies after that, dropped three more shots to end his tournament early.
Bourdy was ahead by one at the halfway mark but could have been clear by more after an eagle on No. 2 and four more birdies in his opening nine. He dropped two shots on the back nine, however.
Home favorites McIlroy and Open champion Darren Clarke both shot 69, with Clarke finishing with a 20-foot putt for birdie to ensure he made the cut for the first time this season, one shot behind McIlroy. Another Northern Irish major winner, Graeme McDowell, was tied with McIlroy at 5 under.
McIlroy was watched by girlfriend Caroline Wozniacki after she made the trip from London after her first-round defeat at Wimbledon.
The No. 2-ranked McIlroy still wasn't at his majestic best, but said he felt some improvement ahead of the weekend -- and next month's Open Championship.
"The last two days have probably been the best ball-striking rounds I've had for a while," he said. "I didn't do much wrong and hopefully I can have a good weekend. I need something around 64 or 65 to get myself into contention, but I've shot low scores here before."