They say breeders "will sell them on to people who are buying the dog as a fad rather than based on an educated decision about what is right for them".
Caroline Kisko, Kennel Club Secretary, said: "We're certainly not pushing any blame away or saying that our breeders are innocent and they are all doing fabulously, but unfortunately the over popularity of these breeds means that there is an opportunity (there) for both puppy farmers and unfortunately for dogs being brought in from Eastern Europe, again with very poor welfare standards.
"The result of those is that they are bred any which way in order to make sure they fulfill a market with no worries about health."
Tory Lord Black of Brentwood has called for the introduction of improved regulations to stop the cruel breeding trend.
"I think we need to look at introducing regulation (that) now exists in some countries to stop the torture breeding of these animals," he said.
"About a half to two thirds of these animals have severe breathing problems and spend their lives struggling for air, often collapsing, often overheating.
"The Scottish fold cat has early onset crippling arthritis because of the genetic changes, many of these animals have difficulty picking up food because the genetic modification means changes to their jaw.
"But sadly, these animals are becoming more and more popular because people think they're cute."
People looking for a new pet are encouraged to visit a rehoming centre instead of looking at designer breeds online.
Dog organisations have also warned of the craze for tiny "teacup puppies" as breeders sell sick dogs with fragile bones and brain problems.
