The Pentagon said there had been a number of US and civilian casualties in what was a "complex" attack.
The attack came after warnings there could be militant attacks, as nations evacuate people ahead of a 31 August deadline.
A spokesman for the Taliban has condemned the twin blasts in the area they say was under the control of the US military.
"The Islamic Emirate strongly condemns the bombing targeting civilians at Kabul airport," Zabihullah Mujahid said in a tweet.
The spokesman added the explosions "took place in an area where US forces are responsible for security".