The bomb went off near a checkpoint at the entrance to the airport complex, which also houses the base of AMISOM- the African Union force fighting Al Qaeda-linked Shebab rebels- as well as a number of foreign diplomatic missions and United Nations offices.
There was no immediate claim of responsibility, but the attack bore all the hallmarks of Shebab, who are fighting to overthrow Somalia's internationally-backed government.
"A car was parked near the airport gate. It detonated. There are casualties," said police official Mohammed Abdi Rahman.
"The airport was busy when the explosion occurred," he added.
Britain's ambassador to Somalia Neil Wigan, whose embassy is also within the high-security airport complex, said on Twitter that he had heard a "major explosion" that sent smoke into the air.
An AFP photographer at the scene saw the burning wreckage of a vehicle and several destroyed shops.