The airport was closed to all civilian air traffic because of the attack, an airport official said. It was unclear if the attack had damaged facilities inside the airport itself.
Kabul police chief Gen. Mohammad Ayub Salangi said at least two insurgents had been killed so far. There were no casualties among police or civilians, he added.
Taliban spokesman Zabiullah Mujahid claimed responsibility for the attack saying the insurgents were targeting NATO.
It was the latest in a series of attacks against the capital since the start of the year. Taliban insurgents have launched intense attacks across the country as Afghan forces take over most security responsibility ahead of most foreign troops' withdrawal next year, more than a decade after the American-led invasion to oust the Taliban regime for sheltering al-Qaida's leadership after the Islamic extremist group launched the Sept. 11, 2001, terrorist attacks in the United States.
The last major attack occurred on May 24, when six suicide bombers attacked a guest house belonging to the United Nations-affiliated International Organization for Migration, killing three people including a police officer, a guard and a civilian. On May 16, a suicide bomber had rammed a car into a NATO convoy killing 15 people, including two American soldiers and four civilian contractors.
Kabul police said in an announcement that attackers wearing suicide vests had occupied two buildings under construction on the west side of the airport and were firing at the military facility, which was quite a distance away. But it was uncertain if they actually managed to hit anything inside the airport.
A statement said there was at least one large explosion at around 4:30 a.m. and a gunbattle began with security forces.
"It first started with a big explosion which we think was a suicide attack. After that a gunbattle started," said Kabul police spokesman Hashmat Stanikzai. He added that at least five insurgents then occupied two buildings, located in a single compound, and started firing rocket-propelled grenades and automatic weapons.
"Sometimes they are shooting from one building, sometimes from other. It is a residential area and the compound has been surrounded by Afghan security forces. The security forces surrounded the buildings and are being careful because it is a residential area," he added.
Afghan army Gen. Murad Ali Murad said police and military forces were exchanging gunfire with insurgents.
"Outside the airport, in the civilian area, there is a tall building under construction and they are shooting at the military side from there. The building is surrounded and there is sporadic shooting. Not enough to damage the military side. We don't know how many terrorist bombers are in there," he said.
The U.S.-led NATO coalition's Joint Command headquarters at the airport runs the day-to-day operations of the nearly 12-year-old war against insurgents. The airport's military side is also used for NATO transport and other aircraft.
"It started just after dawn prayers and I counted about a dozen explosions, mostly RPG fire, coming from the airport," said Emayatullah, who lives next to the airport. Like many Afghans he uses only one name.
The International Assistance Force's Joint Command said it was aware of reports of an attack but had no further details.
Associated Press writer Amir Shah contributed from Kabul.