Round two was halted about 10:30 GMT because of strong winds and it was announced soon after that play would resume at 03:30 GMT on Saturday.
McDowell lies tied fifth on four under par after six holes of his second round, having carded a 68 on Thursday.
The Portrush man posted two birdies and two bogeys on Friday, with George Coetzee leading on nine under.
South African Coetzee was three under for his first eight holes when the players were called to the clubhouse at the Emirates Golf Club.
First-round leader Sergio Garcia is eight under after completing five holes, with fellow Spaniard Nacho Elvira third on six under and American David Lipsky five under.
McDowell posted bogeys at the first and fifth holes on Friday, but picked up shots at the second and third.
Darren Clarke looks set to miss the cut as he is joint 80th on three over after eight holes, while Paul Dunne struggled to a three-over 75 on Friday to lie in a tie for 63rd on one over.
2010 US Open champion McDowell finished joint 28th at the Qatar Masters in Doha last week.
Graeme McDowell lay five shots off the lead when play was abandoned on the second day of the Dubai Desert Classic.