The animal welfare charity said a member of the public spotted the black plastic bucket with a lid on near the entrance to King George V playing fields in Cheltenham on 24 March.
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The puppies, which have been called Mouse, Gus and Olive, were found in a poor state and were rushed to a vet.
The trio are believed to be Shar-Peis and around five weeks old.
The RSPCA said they were suffering from sarcoptic mange – a parasite that causes intense itching and leads to loss of fur.
RSPCA inspector Richard Carr said they look a little bald at the moment but added "we are hopeful they will all make a good recovery."
He said they were being cared for at the RSPCA Cheltenham & East Gloucestershire branch, but there were concerns about their mum and other members of the litter who may also be suffering from mange.
Anyone with any information can call the appeal line on 0300 123 8018.
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