Thick smoke could be seen rising from the area, where there is also a base for African Union (AU) peacekeepers.
Local journalists said one of the bombs was detonated by a suicide attacker near a checkpoint. Another blast hit the airport’s perimeter wall.
Those killed were believed to include security guards at the checkpoint.
The militant Islamist group, al-Shabab, has said it was behind the blasts and its target was the AU force’s headquarters.
The AU mission in Somalia condemned “these senseless attacks that aim to disrupt and cripple the lives of ordinary Somalis”.
Al-Shabab has carried out frequent attacks in Somalia, including in Mogadishu, in an attempt to oust the UN-backed government.
The group, which is allied to al-Qaeda, has been pushed out of most of the main towns it once controlled, but analysts say it remains a potent threat.
It has been increasing its attacks ahead of planned elections in Somalia.
Hotels have been attacked by car bombs and then armed assault teams over the past few weeks.
The large airport area in Mogadishu is a secure “green zone” for UN operations, the AU peacekeeping force and foreign embassies.
The Somali government, with the help of AU forces, is fighting al-Shabab militants in several parts of the country. -BBC