A firefighter told reporters that 13 people are dead and at least 35 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.
Belgian media have said the airport explosions were a “suicide attack”.
An explosion has also taken place at Maelbeek metro station in Brussels centre near the European Union (EU) headquarters.
There are now fears of several coordinated attacks across the city and the entire metro system has been shut down.
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.