Prior to the blasts shots were reportedly fired and onlookers heard Arabic shouting, according to a local news agency.
The cause of the explosions is not yet known but the airport has been evacuated and has been closed to flights. All flights en route to the airport, also known as Zaventem airport, have been diverted.
Belgian broadcaster VRT has said that the airport blasts killed 14 people while in a separate attack 20 people were killed at the metro station in central Brussels. Scores more have been injured.
There are also reports that more explosive devices have been found by firefighters at the airport, the Metro reported.
Passengers were evacuated by armed police.
Footage showed ceiling tiles strewn across floors, along with luggage belts and bags at a check-in desk.
Airport spokesperson Anke Fransen said: "There were two blasts in the departure hall. First aid team are in place for help,” Sky News reported.
Despite reports, it is not thought that American Airlines has been specifically targeted, as the explosion at the airport happened in the west departure hall near to the Brussels Airlines check-in desk and a Starbucks.
American Airlines said all its staff have been accounted for with no reported injuries but that its flight AA751 to Philadelphia had been cancelled.
British Airways said its flight BA389 had departed safely from Brussels but that two other departures to the city this morning, the BA388 and BA392, were cancelled. A decision will be taken later on whether another two flights to Brussels this afternoon will depart. “We will see how things progress,” said a spokesman.
EasyJet has no flights between the UK and Brussels but does have some flights between Brussels and other European airports and expects some disruption and delays today. A spokesperson added: “EasyJet had 14 flights scheduled to arrive and depart Brussels today. Flight EZY2796 from Milan to Brussels has diverted to Maastrict. We recommend to all customers flying to or from Brussels today check the status of their flight on our Flight Tracker page.
“Our thoughts are with those affected by the incident at Brussels Airport.”
A United Airlines flight en route to Brussels from New York this morning was rerouted and has now landed at an unnamed location. The airline has suspended all flights to Brussels. A spokesman added that all its staff in the Belgian capital were safe and accounted for.
Owen Li, a financial services worker from London, told the BBC he was in the affected check-in area 30 minutes before the explosions.
"I was eating my breakfast and then suddenly everyone was running towards us. People didn’t know what to do. They thought it was just an exercise. Some still stayed at the restaurant but slowly people realised it was real."
Meanwhile there has also been another explosion at a station in central Brussels at Maalbeek station, which has closed down the Metro system. TV images reportedly showed smoke billowing out from the station.
One Twitter user, @alxdm captured the image below.
https://twitter.com/alxdm/status/712214397482094592
Heathrow has tightened security in the wake of the blasts. A spokesperson said: “In the light of events in Brussels airport, we are working with the police at Heathrow who are providing a high visibility presence. We expect flights to Brussels Airport to be affected and ask passengers travelling there to check their flight status with their airline.”
Gatwick said it had also increased its security "and patrols around the airport".
Eurostar said it was "running a normal service" to the capital, but advised customers to take extra time for check-in. A spokesperson added that the situation "may change later".
The suspected terror attack comes just days after the arrest of Salah Abdeslam, one of the alleged masterminds behind the devastating November Paris attack, who was captured in Brussels on Friday.
Belgium has now raised its terror threat to level four - its highest level.