The bomber detonated the explosives at a military roadblock near the entrance to Camp Chapman, said Faizullah Ghairat, the Khost city police chief.
Authorities were unable to specify the victims' identities or say if troops were among the casualties.
There was no immediate claim of responsibility, but the Taliban has often targeted Afghan and foreign troops.
"We counted 18 dead and six wounded," said a doctor at a hospital in the city.
Foreign troops including US soldiers are stationed at Camp Champan alongside Afghan soldiers.
A statement from the NATO mission in Afghanistan said it was "aware" of the blast, but did not elaborate.
In 2009, Camp Chapman was the target of a spectacular suicide attack claimed by Al-Qaeda. Seven CIA American officials were killed in the deadliest assault on the US agency since 1983, when eight officers died in an attack on a military base in Beirut.
Camp Chapman is located less than four kilometres (2.4 miles) from the city of Khost, which is near the Pakistani border, a volatile region where the Taliban but also a multitude of armed Islamist groups hold sway.
