Sydney - Authorities are reconsidering a decision not to hunt a great white shark believed to have killed a man at a beach near Perth following a second attack on Tuesday, according to a news report.

A shark attacked a 5-metre dinghy off the same beach and a married couple who radioed for help were rescued by another boat before it sank, ABC Radio said.

It quoted Tony Cappelluti, a spokesman for the local fisheries department, as saying it was not known if it was the same shark that attacked and killed local bank manager Brian Guest, 51, off the same beach at the weekend.

"It's almost impossible to say it's the same shark given the time that's lapsed between Saturday and today," he said.

The dead man's family asked authorities not to kill the shark, saying he loved the sea and accepted the risk of an attack.

"However, in the wake of a second attack it is understood authorities are now considering if the shark should be destroyed," the report said.