Pentagon spokesman John Kirby said one blast occurred near the airport's Abbey Gate and the other close to the nearby Baron Hotel. Two U.S. officials said at least one of the explosions appeared to be from a suicide bombing.
A Taliban official said at least 13 people were killed in the blast, including children, and many Taliban guards were wounded.
A U.S. official told Reuters as many as three U.S. service members were among those hurt, and that U.S. casualty numbers were expected to increase, according to initial information. At least one of the U.S. personnel was seriously hurt, an official said.
The U.S. Embassy in Kabul described "a large explosion" and said there had been reports of gunfire.
The explosion came after the United States and allies urged Afghans to leave the area because of a threat by Islamic State.
A Western diplomat in Kabul said that areas outside the airport gates had been "incredibly crowded" again despite the warnings.