Cats, rabbits, rats and mice are included. So are canaries, budgerigars, tropical fish, terrapins and even worms.
But the legislators are still working on ferrets, which, like many other animals, can carry rabies.
Ben Bradshaw, the animal health minister, said: "When the regulation takes effect next July, rabbits, guinea pigs, gerbils, hamsters and mice from the EU will no longer have to be quarantined for six months when they enter Britain."
Since the passport scheme was introduced in 2000 for dogs and cats, it has been extended to America, Canada and rabies-free islands such as Australia, New Zealand, Japan, Cyprus and Malta. The Department for Environment, Food and Rural Affairs is seeking clarification from Brussels on whether rodents and rabbits will be allowed to travel outside Europe within the scheme.
Dogs and cats must have microchips inserted, be vaccinated against rabies and treated against ticks and tapeworms. The requirements are likely to be different for rodents and rabbits.
Peter Jinman, the president of the British Veterinary Association, said: "There must be active monitoring of pets arriving in Britain. That is no longer so with dogs and cats."
Since the passport scheme was extended to North America, the monitoring has been passive. As a result, for the first time in this country we are recording diseases such as 'brucella canis' and 'leishmania infantum', which can severely affect or kill people.
"We do not want to see the same thing happening with diseases that are in Europe but not so far in Britain."