Boats and helicopters searched the ocean off Sorrento Beach in Perth’s northern suburbs after the late morning attack, but police would not say whether potential remains had been found.
However, police were confident the “swimmer died as a result of the attack”, a statement said.
A witness to the attack, Darrel Haythornthwaite, told Nine Network television that the victim had been swimming with a now traumatised friend.
“I was just shocked. I mean, I feel for his mate. He’s apologised and said ‘I didn’t save my mate’,” Mr Haythornthwaite told Nine. “I said, ‘You couldn’t have done anything’.”
Authorities closed the beach to swimmers (Andrew O’Connor/Australian Broadcasting Corp via AP)
Witnesses reported seeing blood in the water.
Beach walker Jacqui Rapaic told Australian Broadcasting Corp: “My friend said, ‘Jac, I think there’s been a shark attack’, and I looked out to the ocean and I could see a pool of blood.”
Local Ryan Rowe reached the beach soon afterwards and said he was was told by an onlooker “someone had been bitten by a shark”.
“I couldn’t believe there was that much blood (that had) come out of a human body,” Mr Rowe told The West Australian newspaper.
Authorities closed the beach to swimmers. Witnesses described the swimmer as a man in his 60s but police did not identify him further.
On Monday, a surfer sustained a serious leg wound in a shark attack near Geraldton, about 250 miles north of Perth.
There have been five fatal shark attacks in Australia this year.
Three spearfishing divers were killed by sharks off the Australian coast in four weeks over May and June.
In January, a 12-year-old boy died in a hospital days after he was mauled by a bull shark in Sydney Harbour.
Australia has averaged between two and three fatal shark attacks a year since 2000, according to the Australian Shark Incident Database, a partnership of the Taronga Conservation Society Australia, Flinders University and the New South Wales state government.