There were at least three explosions followed by gunfire near the airport, CNN reported.
Khaama Press reported clashes continued between the attackers and Afghan security forces and quoted security officials that all flights at the airport had been suspended.
The report also quoted another security official as saying there was another explosion inside the capital prior to the incident near the airport.
Other details were not immediately available.
Duggar sisters unveil Christian dating rules in new book
Attkisson leaves CBS News, reportedly over network's 'liberal bias'