9 September 2010 Last updated at 14:27 GMT
Unconfirmed reports say AU soldiers and insurgents were killed in the battle.
The hard-line Islamist group al-Shabab, which is fighting the government for control of the country, has reportedly said it carried out the attack.
Heavily fortified The officials said several insurgents were killed, along with a Somali man working with a private airline.
Mogadishu's airport, of huge strategic importance, is one of the few parts of the capital controlled by forces from the AU and the government.
Mohammed Abdi, a local shopkeeper, told Reuters news agency that he had witnessed two explosions at the airport's heavily fortified main entrance.
"The car powerfully rammed an Amisom [African Union Mission in Somalia] troops' post at the airport's gate," he said.
He said he had seen eight bodies, including several AU soldiers.
Yusuf Bashir, another local shopkeeper, told Spanish news agency Efe that eight civilians had been killed, and he said he had seen several AU soldiers injured.
As well as the car bomb, two suicide attackers had arrived at the airport on foot and detonated their explosives, said Mr Bashir.
A Somali military official told the Associated Press that two women had been killed while begging outside the airport's entrance.
In recent weeks, al-Shabab has stepped up the intensity of its insurgency against government forces and AU peacekeepers.
On Wednesday, 11 people were killed in clashes between al-Shabab fighters and Somali troops.
In the worst single attack, gunmen from the Islamist group stormed a hotel last month and killed at least 32 people, including six MPs.
The government had said that it expected a surge in rebel activity as the Muslim holy month of Ramadan drew to a close.
Al-Shabab is fighting to overthrow the government and install its own radical interpretation of Islamic law.
Al-Shabab and other Islamist rebels control large parts of the country, while the government holds only small parts of Mogadishu.