McIlroy and Bjorn posted bogey-free rounds to get to 13-under 131 after 36 holes. First-round leader Rafael Cabrera-Bello had a 69 Friday and is alone in third at 12-under.
Gregory Bourdy, Scott Jamison and Martin Kaymer are tied for fourth at 11-under. Kaymer's round of 67 included a hole-in-one at the 186-yard seventh hole. Bourdy also shot 67 and Jamison had a 68.
McIlroy, who played the back nine of the Emirates Golf Club in 6-under Thursday, opened on that side Friday and had four more birdies. He had three birdies on the front nine, including one at the third hole that left him 4-under on the par 5s Friday.
Bjorn was 1-under after eight holes but had four birdies over Nos. 9-13 and closed with back-to-back birdies for his share of the lead.
Cabrera-Bello was three back of the leaders after a bogey at 15 but closed the gap with birdies at 16 and 17.
2014: The Year in Music [PHOTOS]