Updated: The WHO has since removed this information from their "Myth Busters" page, likely in light of the case(s) cited in trond hansen's answer.
There is currently no evidence that domestic cats or dogs are affected by COVID-19:
Can pets at home spread the new coronavirus (2019-nCoV)?
At present, there is no evidence that companion animals/pets such as dogs or cats can be infected with the new coronavirus. However, it is always a good idea to wash your hands with soap and water after contact with pets. This protects you against various common bacteria such as E.coli and Salmonella that can pass between pets and humans.
Source: World Health Organization