Bjorn, 41, hit a sparkling four-under-par 67 at par-71 Sentosa Golf Club to give himself a great chance of his first win of the year, especially after weather delays left the event at risk of being cut to 54 holes.
McIlroy, needing a top-three finish to clinch the European order of merit, as he seeks to win the money lists on both sides of the Atlantic, shot his second successive round of 70, leaving him seven shots back from the leader.
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