The black, semi-long haired cat was found in a blue recycling bin at the side of the road near the children's nursery in Stonygate on Thursday (March 25).
The person who found her took her home and alerted the animal welfare charity, who sent Inspector Alison Fletcher to the scene to help.
Alison said she found the cat "trembling constantly" and unable to stand. She rushed the weakened cat to the charity's animal hospital in Salford where it collapsed on the vet's table.
The cat was given medicine and monitored for several hours, with the vet hoping she might pull through, but sadly, the decision was made to put her to sleep to end her suffering.
The vet believes her weakened state was due to a long term illness and the RSPCA suspect her owner had dumped her in the bin "like worthless rubbish" to avoid paying for treatment.
Now the RSPCA inspector is keen to trace the owner of the cat, which was not micro-chipped, and find the person responsible for the "callous" incident.
Alison said: "This poor cat was in a terrible state - she was trembling and weak and was obviously very frightened and in need of help.
"It is hard to comprehend how anyone can do this to a pet. It appears they decided to put her in the bin like worthless rubbish in her hour of need rather than seek veterinary treatment for her.
"I am keen to find the person who owned this cat to find out what happened to her and am appealing for anyone who recognises her and knows who she belongs to to get in touch.
"I am also keen to find anyone who may have witnessed the cat being placed callously in the bin."
Anyone with information can call the RSPCA’s appeals line on 0300 123 8018.