The Dane leads after three holes on his third round which he is due to complete tomorrow after play was suspended today due to darkness, as the players continue to battle against the elements.
Rain delays had disrupted day two with Bjorn one of a number of players who did not play at all on Friday.
Having added a 67 to his first round 66 Bjorn returned to the course to start his third round but had only parred the first three holes when a halt was called, with the players due to resume round three at Sentosa at 0730 on Sunday morning.
Bjorn saw his first round lead taken away by England's Chris Wood and South Africa's George Coetzee but reeled them in to reclaim the lead as round two progressed.
Bjorn picked up the last of his four birdies on the 14th and finished with four pars to complete his second round on nine under.
Meanwhile, world number one McIlory is on four under after the Northern Irishman made up for shooting 70 for both of his first two rounds by picking up two shots in the first nine of his third round.
McIlroy said: "Two under through nine was a pretty decent effort, but I probably need a few more birdies tomorrow morning to get myself within a few shots of the lead going into the fourth round."
Bjorn, who then opened his third round with three pars before a halt was called to play, was in good spirits as he looked for his first win of the year.
"This course can very quickly bite you, especially if you don't drive the ball well," Bjorn told the European Tour website after completing his second round.
"The numbers can run up on this golf course, so you've got to stay with what you're doing.
"I've enjoyed these two days, and it's nice to play some good stuff but there's still a long way to go in this golf tournament.
"I've got two good rounds in the bank but there's a way to go."
Coetzee and Wood, both looking for a maiden victory on the European Tour, ended the second round level with the former adding a best-of-the-week 63 to a 72 and Wood firing 65 after an opening 70.
The Englishman moved ahead into second overall with a birdie on the second, putting him one behind Bjorn.
Thailand's Chapchai Nirat had led after an opening 65 but a second round 73 hurt his chances only four two birdies in the four holes he completed of round three to leave him joint fourth on six under with Italian Francesco Molinari.
A seven-strong group of players on five under included Australia's Adam Scot, England's Simon Dyson and Italian Matteo Manassero.