Under the proposals, people buying or adopting a dog would deal directly with a breeder or rehousing centre.
Environment Secretary Michael Gove said the plan would be examined as part of a new package of measures aimed at driving up welfare standards.
Gove said he wanted to introduce a ban on third-party sales of puppies "to do everything we can to make sure the nation's much loved pets get the right start in life".
Under the new rules, breeders or sellers of dogs must be licensed and will not be able to sell puppies and kittens under eight weeks old, the BBC reported.
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Puppies must also be shown alongside their mother before a sale is made, and - amid concern over online sales - purchases must be completed in the presence of the new owner.
Some 800,000 dogs are sold in the country every year, mostly through breeders.
Last year, the Department for Environment, Food and Rural Affairs suggested a ban on third-party sales could lead to the creation of an illegal market, The Independent reported.
Paula Boyden, veterinary director for the Dogs Trust, said: "If a ban was introduced now, puppy farmers could exploit loopholes such as setting themselves up as unregulated re-homing centres or sanctuaries.
"Licensing and inspection of dog breeders and sellers must also be stronger to ensure that everyone involved in the trade is on the radar of local authorities."
The deputy chief executive of the Royal Society for the Prevention of Cruelty to Animals Chris Wainwright said: "We have always said that an end to third party sales alone would not be enough to end the puppy trade crisis, and we are pleased that this is being looked at alongside enhanced licensing conditions for breeders."
He said: "We've also found tiny puppies kept in buckets amongst the dead bodies of their siblings.
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