"The drones have disappeared and the airport is open again," Deputy Police Inspector Jakob Hansen told reporters.
"We didn't take the drones down," he added.
The airport was closed for several hours before reopening early Tuesday, causing numerous delays and travel disruptions to 20,000 passengers, airport officials said.
Hansen said police were cooperating with the Danish military and intelligence service to find out where the drones had come from.
The unidentified drones were observed near the airport, causing around 15 flights to be diverted, police and airport officials told news agency AFP.
"The airspace over Copenhagen airport has been closed since 8:30pm due to two to three unidentified drones. No aircraft can take off or land," airport spokeswoman Lise Agerley Kurstein said late on Monday.
She said around 15 flights had been diverted to other airports.
Copenhagen Police meanwhile said "three or four big drones" had been observed flying over the airport.
"They are still flying back and forth, coming and going," duty officer Anette Ostenfeldt told AFP at 10:45pm, adding that police were at the airport investigating.
She could not say if the drones were military or civilian devices.
The airport has now reopened but remains disrupted this morning with possible flight delays. Passengers are advised to stay updated via the airport’s website or their airlines.
Hansen also said police were also working with colleagues in Oslo after drone sightings in the Norwegian capital also caused the airport to close for several hours.
"We had two different drone sightings," Oslo airport spokeswoman Monica Fasting told AFP.
"We reopened the airport around 3:15am," she said.
Flights were diverted to nearby destinations during the closures, and officials at both airports said they expected some delays and disruptions to continue on Tuesday.
The incidents came after the governments of Poland, Estonia and Romania accused Russia of violating their airspace this month, allegations that Moscow has brushed off.
Danish police said on Tuesday morning that they did not know who was responsible for flying the drones over Copenhagen Airport the previous evening, but that they appeared to have been knowledgeable.
"The number, size, flight patterns, time over the airport. All this together ... indicates that it is a capable actor. Which capable actor, I do not know," police inspector Jens Jespersen told reporters.
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