Deputy Interior Minister Gen. Mohammad Ayoub Salangi said security personnel were sent to the airport where a firefight was under way in the early part of the day.
A number of explosions were heard throughout the capital, as well as gunfire, Khaama Press reported.
The militants took position in a building currently under construction near the airport.
An interior ministry official said the insurgents were armed with light and heavy weaponry, including rocket-propelled grenade launchers.
The airport was closed down to all traffic.
The violence comes one day after 89 people were killed by a roadside bomb in Afghanistan's eastern Paktika province.
2014: The Year in Fashion [PHOTOS]
Syrian Al Qaida group executes Lebanese soldier