A spokesman for NATO's Resolute Support Mission, which trains and advises Afghan forces, did not immediately confirm the number and severity of the casualties, or the type of attack.
"We are working to gather additional information as quickly as possible and will release more details as appropriate," it said in a statement.
The airport in Kandahar is home to a major military base for international troops helping Afghan security forces.
The combined U.S. and NATO troop contingent currently in Afghanistan is about 13,500. The White House is deciding whether to send about 4,000 or more U.S. soldiers to Afghanistan in an attempt to stem Taliban gains.