A planning application to City of York Council says the site in Landing Lane, Holgate, is "no longer fit for purpose".
The current facility is made up of 17 buildings, from which staff work to rehome stray animals. So far this year, the shelter has rehomed almost 400 dogs, cats and small animals.
RSPCA bosses held a meeting with local residents on January 30 to discuss plans to demolish all the buildings, and say they received no complaints.
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