The authority said an inspection would be conducted by a licensing officer and a suitable, qualified person such as a vet or experienced zoo keeper as part of the application process.
Owners of chimpanzees, baboons, gorillas and certain species will also need to apply for a dangerous wild animal licence costing £430.
Councillor Lyn Healing, cabinet member for communities and public protection said: “These regulations are vital for animal welfare, and we urge any residents who own primates to get in touch with the licensing team as soon as possible.
“Together we can safeguard animal wellbeing across our region.”