(CNN) -- Rory McIlroy trailed leaders Justin Rose and Tom Gillis by just a shot as he stepped up his bid to become world No.1 at the halfway stage of the Honda Classic Friday.
The 22-year-old from Northern Ireland birdied the par-five 18th at PGA National in Florida to close with a second round three-under-par 67 for seven-under 133.
American journeyman Gillis and England's Rose shared top spot on 132 going into the third round.
Rose, who shot a second straight 66, looked set to lead outright but successive bogeys on his back nine saw him fall back into a share with Gillis, who carded a superb 64.
McIlroy kept up his dogged pursuit, carding three birdies against a single bogey on the tough back nine, to stay firmly in contention for the win he needs to guarantee he will leapfrog Luke Donald at the top of the rankings.
"I felt like I played another really solid round," he told gathered reporters.
"I had a couple of chances on the front nine and didn't take them, but I was
just trying to stay as patient as possible."
Round of the day came from American rookie Brian Harman, a remarkable course record 61, shattering the previous best by three shots.
Harman had the chance to break the magical 60 barrier with his bunker shot on the long 18th. The eagle attempt ended five-feet from the cup and he missed the short birdie putt for a 60.
It left him on six-under par overall after a first day 73.
Earlier, Tiger Woods salvaged his round and made sure he made the cut with back to back birdies on his final two holes, the eighth and ninth, to finish with a two-under 68.
Woods is on one-under 139 after two rounds and still unable to build on sporadic strong play.
"It was a grind," Woods told the official PGA Tour website.
"I got it going, lost it, got it going, lost it and got it going.
"I didn't have it, but I scored and that's something I can take out of it."
Playing partner and world No.3 Lee Westwood tied Woods on 139 after a second round 69.