It comes a few weeks after the country reported its first Covid-19 case in an animal - a kitten.
Only pets which show symptoms like fever or breathing difficulties after being exposed to human carriers of the virus will be tested.
The pet must quarantine at home if it tests positive for the virus.
It is not necessary for the pet to be sent into an isolation facility as there is no evidence Covid-19 can be spread between humans and pets, said Park Yoo-mi, a disease control official at a virtual press briefing, according to a Yonhap report.
But if the pet's owners are hospitalised with Covid-19, or are too ill or too old to take care of it, the pet will be taken into quarantine at a city-run facility.
In South Korea, human patients with Covid-19 are generally put in quarantine facilities if they do not need hospital treatment.
Ms Park reminded residents to keep their pets "at least two metres away from people and other pets when walking them".