Watch Marina Jenkins' report
The pair were around six weeks old when they were found on Monday 18 July.
They have been named Duck and Goose, and have been placed in kennels by the council's dog warden who says they are doing well despite their 'ordeal'.
In an online post Wiltshire Council said: "Both pups are doing well and are currently undergoing care and assessment at our holding kennels.
"We will endeavour to keep updating our followers on their progress as and when we can."
There have already been many people offering to foster and adopt the spaniel-type puppies.
The council's dog warden has thanked the public reassuring that the puppies are safe and well.
Wiltshire Council said the circumstances behind the incident are being investigated and they are appealing for anyone with information to get in touch.
They said: "We are asking the public for help to try and locate the owner and breeder. Do you know of anyone who has had some puppies recently and they seem to have disappeared suddenly?"