A firefighter told reporters that 11 people are dead and at least 20 people are injured.
A Belgian news agency also reported that witnesses had said they heard gunshots and shouts of Arabic at the time of the explosions fuelling fears that this may have been an attack inspired by islamic State (ISIS) terrorists.
An explosion has reportedly taken place at Maelbeek metro station in Brussels centre.
The large explosions have taken out glass windows at the entrance and much of the airport's ceiling has collapsed.
Other explosive devices have been discovered at the airport, firefighters have told journalists at the scene.
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Injured passengers at Brussels airport
The cause of the two blasts, which occurred at around 8am this morning at an American airline check-in desk, are unknown at this stage.
There are unconformed reports that a bomb had been placed next to a check-in desk.
Belgian Prime Minister Charles Michel has posted on Twitter: "We are following the situation minute by minute. The absolute priority is for the victims and people who were at the airport."
Belgian authorities have closed the metro in the capital of Brussels following the explosions, according to Belgian newspaper Le Soir.
The blasts occurred four days after the arrest in Brussels of Salah Abdeslam, suspected of being involved in November's Paris massacre that left 130 people daed and more than 300 injured.
Abdeslam's accomplice Najim Laacheaoui, who was in hiding with the Paris terrorist in the Brussels district of Molenbeek, is still on the run.
Belgian police had been on alert for any reprisal action.
Social media showed pictures of smoke rising from the departure hall where all windows had been shattered by the blast. Passengers were seen running away down a slipway from the departure lounge.
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It is unknown how many are injured
Rudi Vervoot, the president of the Brussels region, said: "There has been an explosion in the airport. An emergency plan will certainly be activated."
A witness said: "I was in the check-in queue and I heard an explosion. I saw smoke, I saw people panicking and running towards the exit. There was a second explosion much closer to me.
"Everyone left the airport in panic, most people left their baggage."