Thousands of people have gathered at the airport over the past 12 days hoping to be evacuated after the Taliban seized power.
At least 60 Afghans have been killed and 140 other people hurt in the attacks.
At least 12 US service members were killed, including 11 Marines and one Navy medic, while 15 were injured, the Pentagon said.
Marine Corps General Kenneth McKenzie, commander of US Central Command, said: “At this time we know that 12 US service members have been killed in the attack, and 15 more service members have been injured.
“A number of Afghan civilians were also killed and injured in the attack.”
Other blasts were heard in Kabul in the hours after the earlier two.
ISIS claimed responsibility for the attacks, sharing a picture online of one of the suicide bombers.
Pentagon spokesman John Kirby said one explosion occurred near buses lined up outside Abbey Gate and the other close to the nearby Baron Hotel.
A source told Fox News that the attacks in Kabul may be an “ongoing event” – with more violence to come.
One of the bombers reportedly hit people standing in a wastewater canal – sending bodies flying into the water.
Bloodied survivors were raced from the scene in wheelbarrows, according to local TV news channels.