Gatwick's runway has been shut since Wednesday night, as devices have been repeatedly flying over the airfield.
Sussex Police said it was not terror-related but a "deliberate act" of disruption, using "industrial specification" drones.
About 110,000 passengers on 760 flights were due to fly on Thursday. Disruption could last "several days".
An airline source told the BBC flights were currently cancelled until at least 19:00 GMT.
The airport advised that the runway would not open "until it was safe to do so".
Defence Secretary Gavin Williamson has confirmed the Army has been called in to support Sussex Police.
He said: "The armed forces have a range of unique capabilities and this isn't something we would usually deploy but we are there to assist and do everything we can so that they are in a position to open the airport at the earliest opportunity."
Those due to travel have been told to check the status of their flight, while Easyjet told its passengers not to go to Gatwick if their flights have been cancelled.