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World number 24 Reed, seeking his first major title, was 15-under through 15 holes after four birdies, two bogeys and two eagles.
McIlroy eagled the par-5 eighth to match Reed briefly for the lead at 9-under, but could only add a birdie at 15 to stand on 10-under with two holes remaining.
Meanwhile his move lit a fire under Reed, best known for his fiery play on the US Ryder Cup team.
American Rickie Fowler was third on 9-under 207 with Spain's Jon Rahm fourth on 208 after each fired the low round of the week of 7-under 65.
Sweden's Henrik Stenson was fifth at 7-under with two holes remaining while two-time Masters winner Bubba Watson and England's Tommy Fleetwood were in the clubhouse on 6-under 210.