Muhammad Ali Johar, a spokesman for the regional government, said militants had opened fire on the bus on Saturday evening and the wounded had been taken to a local hospital.
No group immediately claimed responsibility for the attack and the motive for the shooting was not clear.
Chilas lies in the mountainous region of Gilgit Baltistan, near the province of Khyber Pakhtunkhwa, where attacks have been rising in recent years, including some claimed by the Pakistani Taliban, or Tehreek-e-Taliban Pakistan (TTP).
Chilas is a popular stopping point for tourists and is also near a China-backed dam under construction.
(Reporting by Mushtaq Ali in Peshawar; Editing by Kevin Liffey)