Beloved pet 'Scampi', who lived in the South Elmsall area, was picked up by a couple after wandering away from his owners on a walk, a ward councillor claimed.
Coun Michelle Collins said that the 18 year-old was handed over to the council's dog wardens, who then allegedly told the finders that he would be taken to a vet, and then onto a kennels if he could not be reunited with his family.
However, Scampi was instead put to sleep at the vet's practice, even though his owners were said to be actively looking for him.
Coun Collins said she wanted to help Scampi's family "piece together" the circumstances that led to his death.
She told a full council meeting: "The owners had contacted the council and the dog wardens to report him missing, but they weren't told of his fate until after he was put to sleep.
"That dog has been an integral part of that family, so you can understand the upset they're feeling at this time.
"They were unable get Scampi's body back for three days and they then had to pay for the unauthorised euthanasia of their dog.
"I understand dog wardens have difficult decisions to make, but there does seem to be conflicting information about what happened that day."
Coun Maureen Cummings, portfolio holder for communities, confirmed she was aware of the incident and that an investigation had started.
She said: "I'm getting regular updates on this.
"I'm a dog lover myself, so I know how the people will be feeling about this."