The $6,600 dollar reward is the latest twist in a high seas drama which began five weeks ago when the live cargo of sheep was rejected by two Middle Eastern countries due to health fears.
Campaigners say more than 4000 animals in the original shipment of 57,000 sheep have died as the ship MV Cormo Express - cruelly nicknamed by critics the "sheep of death".
The group Animal Liberation called on Agriculture Minister Warren Truss to ensure the remaining sheep were slaughtered humanely.
"We must immediately find this ship's exact whereabouts and we demand that Truss immediately mobilise all necessary resources to be delivered to it to have all animals humanely euthanased before they begin to starve and many more die from heat stress," an Animal Liberation statement said.
"It will require adequate experienced stockmen, guns and charges to be sent to the ship to ensure the deed be done as swiftly as possible," it said.
Saudi Arabia rejected the sheep delivery five weeks ago, claiming that an unacceptable number of the animals were suffering from the disease scabby mouth ... a claim the Australian vet on board the ship denied.
Australia responded by suspending its $129 million dollar a year livestock trade with Saudi Arabia.
The sheep were then rejected by a second Middle Eastern country. The government has insisted the shipment is the responsibility of a Saudi importer who purchased the sheep.
The Senate on Wednesday passed a motion demanding the government provide updated details on Thursday of how many of the sheep had died aboard the ship so far.