Video from the scene shows people injured in the explosions arriving at the hospital.
The total number of dead and injured was not immediately confirmed but the Pentagon said earlier three U.S. marines were among the injured.
A Taliban spokesman told NBC News a total of 50 people were killed or injured including children. The sources described it as a suicide attack.
The bombings throw evacuation efforts from the airport into more turmoil but airlifts from the international airport continued after the explosions.
It was immediately unclear who or what was behind the bombings.
But two U.S. intelligence officials said the assumption is that the attack was caused by the ISIS Afghan affiliate, ISIS-K.
Pentagon spokesman John Kirby said on Twitter that the initial blast outside the airport’s Abbey Gate was the “result of a complex attack.” Kirby said at least one other explosion took place near a hotel that is close to the Abbey Gate.