The attack began about 2 a.m. local time (5:30 p.m. EDT Friday) and lasted about 90 minutes, Khaama Press reported.
At least two NATO troops and 16 militants died in fighting at the Shorab airbase in Helmand province where the prince is based, said Daud Ahmadi, a spokesman for the provincial governor.
A Taliban spokesman said at least 10 coalition soldiers were killed.
Maj. Adam Wojack, a spokesman for the International Security Assistance Force in Kabul, said both British and United States forces were involved in the fighting, The Daily Telegraph reported.
At least two U.S. Marines died. Wojack said 18 attackers were killed and another was wounded and receiving medical attention.
Taliban fighters broke through an outer fence and attacked the airfield inside the 13-square-mile base. Aircraft, hangars and other buildings were severely damaged.
The base, comprised of the British Camp Bastion and the U.S. Camp Leatherneck, houses more than 21,000 people.
Prince Harry, who arrived in Afghanistan 10 days ago in his second deployment as a helicopter pilot, is not believed to have been involved in the fighting. He is to be deployed in Afghanistan for four months.