The woman, 60, was fatally mauled one kilometre (half a mile) offshore from Perth on Sunday, by a shark described by witnesses as bigger than their 5.3-metre (17.4-foot) boat.
A surfer also succumbed to his injuries on Friday after his leg was bitten off by a shark along the same coastline last week.
The West Australian fisheries department said it would deploy "capture gear" to haul in what they believe is a great white shark that poses "a serious threat to public safety".
Woman killed by shark 'bigger than boat' in Australia
Fisheries official Tony Cappelluti said if a shark matching the description of the one that attacked the diver was captured, it was "highly likely we will take the decision to destroy it in the interests of public safety".
"We have two methods... we can either let it expire on the line or we can use a firearm to destroy it," he said. "In this instance today, we will be using a firearm."
Animal rights group Humane Society International expressed "extreme concern" at the government's plans.
"Given that science tells us that drum lines are an unnecessary and lethal option for shark management, the government's response can only be described as revenge killing," campaign director Michael Kennedy said in a statement.
Second suspected shark attack death in Australia
Western Australian Premier Colin Barnett said his state, which has now suffered 14 shark attack fatalities since 2000, was in shock at the latest deaths.
"I don't think there's any doubt that the number of shark attacks and fatalities from shark attack has increased quite significantly over recent years," he told reporters in Perth.
Barnett said a number of measures were deployed, including the aerial patrols, to alert swimmers to sharks.
But the catching and killing of the predators has long proved controversial.
A previous trial involving baited hooks attached to floating drum lines to snare sharks was abandoned after objections from conservationists and marine scientists.
Barnett said, however, that the government would always err on the side of public safety and reserved the right to destroy a shark if it posed an imminent threat.
Last Wednesday, a 4.2-metre shark was caught near to where surfer Ben Gerring lost his leg off Falcon Beach, 115 kilometres south of Sunday's attack near Mindarie, but it is unclear if it was the animal that attacked him.
Surfer dies after losing leg in Australian shark attack
Deadly attacks are relatively rare, according to the International Shark Attack File, which in February reported only six fatalities worldwide last year, including one in Australia.
In Sunday's incident, the woman's dive partner told police they were in the water together when he felt something brush past him.
He spun around but did not find anything and decided to head to the surface where he "saw a commotion".
The man and his rescuers then managed to retrieve the woman, but she died from her injuries, police said.
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