The council hoped to find the bull mastiff overnight. Animal control would be on the lookout, Mr May said, and traps had been laid in the area where the owner lived and where the attacks had taken place.
When both the dogs had been found, the council would determine their fate.
"It depends on the response from the owners," he said.
"We'll look at what we do, whether that be putting them down, fining the owners, getting them to write letters of apology to the families whose pets were killed."
The owner had been taking care of the dogs, but was unaware they had been roaming at night.
Oamaru woman Tracey Taylor, whose family lost a pet rabbit in the recent maulings, said it was a relief one dog had been found.
"We can sleep easy at night knowing we can have the other rabbit in a cage outside at night and it will be safe," she said. "Hopefully the other will be found before long but it's certainly a relief they've found one, especially knowing how many animals were killed."