Stallings birdied three of the last four holes to stand on six-under par 66 after 18 holes, one stroke ahead of fellow Americans Spencer Levin and Erik Compton and three clear of the rest of the star-studded field.
At the Muirfield Village layout designed by Jack Nicklaus, Woods shared 11th on 70 with a group that included Australian Adam Scott and three-time major winner Ernie Els.
World No. 1 Donald of England and McIlroy, the Northern Ireland prodigy who will defend his US Open title in two weeks at The Olympic Course in San Francisco, were another stroke adrift in a group on 71.
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