The official said many Taliban guards were wounded.
A U.S. official said U.S. service members were among the wounded, adding he was citing an initial report and cautioning that it could change. He said there were casualties but did not know how many or of what nationality.
A video posted on social media shows injured people being brought to a hospital in Kabul.
Thousands of people have been gathering outside the airport in recent days. Western troops are racing to evacuate foreigners and Afghans who helped Western countries during the 20-year war against the Taliban, and to get out themselves by an Aug. 31 deadline.