A member of the public alerted police shortly after 1600 GMT on Sunday after stumbling across the remains while walking in Anmer, east England, according to a statement released by Norfolk Constabulary.
Police sealed off the area and conducted a detailed search, the statement added.
The royal family spent Christmas and New Year at the 19,770-acre (8,000-hectare) estate in Norfolk, although celebrations were interrupted when Prince Philip, husband of Queen Elizabeth II, was airlifted to hospital after suffering a blocked coronary artery.
Philip joined the family for a New Year's Day service at the estate's St. Mary Magdalene Church on Sunday, his first public appearance since undergoing surgery.
However, the prince missed the estate's Boxing Day shoot held last Monday.
Sandringham House, much loved by the queen, has been the private home of four generations of British monarchs, attracted by its prime location for shooting.
King George V, the queen's grandfather, famously called "dear old Sandringham," the "place I love better than anywhere in the world."
The queen often spends Christmas at the estate, and makes it her official base until February. The house is opened to the public when the royals are not in residence.
Around half of the estate is let to farm tenants, with much of the remainder used for forestry. The estate also contains two studs, a fruit farm and a country park.
The queen's Buckingham Palace office refused to comment, adding it was "a matter for Norfolk police".