Mr Paphitis, whose other retail interests include the Ryman stationery chain and Boux Avenue lingerie retailer, bought Robert Dyas through his company Gladys Emmanuel Limited.
"The idea really is to provide a beautiful, comfortable, quiet lounge where people can come if they'd like to and sit amongst the cats, " Ms Dyas told BBC News.
Ms Dyas, whose six "sociable" cats would be in the cafe, said pet therapy is acknowledged by many people as beneficial and can help to reduce people's anxiety and stress.