Peter Kurmann, 33, was attacked Saturday in Geographe Bay. The remains reported Friday were on Preston Beach nearby, PerthNow reported.
Sgt. Gerry Cassidy, a police spokesman, said the remains would not be positively identified until next week after a forensic examination in a Perth laboratory.
Kurmann was diving with his brother, who escaped without injury. He is believed to have been attacked by a 12-foot great white shark.
Kurmann's wife, Debra, a member of the local council in Busselton, called him "an amazing father and husband" in a death notice. The couple have two young sons.
Two people were killed in September and October in shark attacks in Western Australia, and a third disappeared while swimming in what is believed to be another fatality, authorities said.
Kurmann was the first person to die in a shark attack in the area in 2012.