Italy's National Institute for Environmental Protection and Research regulations call for bears that attack humans to be euthanised.
After the attack, Trentino governor Maurizio Fugatti signed a cull order allowing for the capture and killing of the bear.
To make sure they kill the right bear, authorities are trying to identify the bear's DNA garnered from saliva and fur left in the claw and bite wounds and on the father's and son's clothing.
There have been a number of bear attacks in the region in recent years, and local authorities have a database of bear DNA collected from faeces, fur and saliva.
Surveillance cameras are used to match the DNA to the animals.
The Italian animal rights groups Animalisti Italiani and the World Wide Fund for Nature have demanded a full investigation before the death is carried out.
In particular, they want to find out if the father or son did anything to taunt the bear, which the men deny.
More than 17,000 people have signed a World Wide Fund for Nature petition to save the bear.
The Italian environment minister Sergio Costa is also opposed to the death sentence. He suggested that the bear may have been a female protecting her cubs.
"Only after collecting certain scientific information on the animal involved in the accident with the two citizens we will be able to evaluate technical solutions that, in my opinion, must not result in killing the animal," Sergio Costa said.