Scotland's animal welfare charity was alerted on Sunday, July 13 after a member of the public spotted the male Soay lamb being chased by a fox into the garden in the Skerray Quadrant area of Milton.
The lamb is now in the care of the Scottish SPCA's Edinburgh and Lothians centre in Balerno, where staff have nicknamed him Shamus.
Inspector Jennifer Connolly said, "Shamus is a native sheep from the Isle of Soay which is classed as a rare breed, so we're pretty certain someone will be missing him.
"We've no idea how he ended up where he did as there are no sheep farms close-by.
"Unfortunately he doesn't have an identification tag or a chip so we're appealing to anyone who recognises him to get in touch.
"It's likely he hasn't travelled too far so we're hoping someone will realise he's missing and come forward. If not we'll find him a new home."
Anyone with information is being urged to contact the Scottish SPCA animal helpline on 03000 999 999.