In a statement, Islamic State claimed responsibility and said one of its suicide bombers had targeted "translators and collaborators with the American army."
Marine Corps General Frank McKenzie, head of the U.S. military's Central Command, told a news briefing the explosions were followed by gunfight. McKenzie said the threat from Islamic State persists alongside "other active threat streams."
"We believe it is their desire to continue these attacks and we expect those attacks to continue - and we're doing everything we can to be prepared," McKenzie said.
McKenzie added that future potential attacks could include rockets being fired at the airport or car bombs attempting to get in. McKenzie said he saw nothing that would convince him that Taliban forces had let the attack take place.
U.S. officials said one bomb detonated near the airport's Abbey Gate and the other was close to the nearby Baron Hotel.
RACE AGAINST TIME
A massive airlift of U.S. and other foreign nationals and their families as well as some Afghans has been under way since the day before Taliban forces captured Kabul on Aug. 15, capping a swift advance across the country as American and allied troops withdrew
Afghan health officials were quoted as saying 60 civilians died, but it was not clear whether that was a complete count. Video uploaded by Afghan journalists showed dozens of bodies and wounded victims strewn around a canal on the edge of the airport.
U.S. officials have said there are about 5,200 American troops providing airport security. The attacks came after the United States and allies had urged Afghans to leave the area around the airport because of a threat by Islamic State.
The U.S. Embassy in Kabul a day earlier had advised Americans to avoid traveling to the airport and said those already at the gates should leave immediately, citing unspecified "security threats."