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.
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 September 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 Organisation for Migration, killing three people including a police officer, a guard and a civilian.
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 am (local time) 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.
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