“Human remains” were discovered in the grounds of the £4 million Gogar Mount House, off Gogarstone Road, last weekend.
An extensive search then uncovered other body parts on the boundary with Gogarburn Golf Club.
Forensics specialists worked at the scene all week and a gazebo was set up by the ninth fairway as a police base as investigations progressed.
The forensic team left the location – which is close to the A8 Glasgow Road – at lunchtime yesterday after recovering the remains, which are said to include a skull.
It is understood that police divers have been searching large ponds on the boundary of the golf course.
Work is now under way to establish the identity of the person, and the circumstances surrounding their death.
Forensic anthropologists from Dundee University, which is a leading specialist in analysing human remains, are assisting police.
The results of the examination are not expected until the end of next week.
A Police Scotland spokesman said that inquiries were ongoing and insisted that it would be inappropriate for families of missing people to be contacted until more was known about the circumstances.
He said that further updates would be provided when available, and that no further comment would be made about the sex of the person.
No-one from the 12-hole golf club, which neighbours the RBS headquarters at Gogarburn, was available for comment. However, it is understood that it has been opened for golfers to play a short course.
The “top field” – including the seventh, eighth and ninth holes – has been closed to allow officers to continue their inquiries.
There are a number of houses within the grounds of the six-bedroom Gogar Mount House, which was sold in 2006.
One resident, who lives nearby, reportedly said: “The police have been searching the burn which runs behind my garden. Someone at the golf club told me skeletal remains had been found. It’s a bit worrying, whatever has happened, so close to home.”
The grisly find comes just weeks after police identified human remains uncovered in West Lothian as belonging to notorious gangster Martin Hamilton.
The 53-year-old was a gangland enforcer who was infamous for a string of sadistic crimes including torturing and raping teenagers.
His remains were found by a dog walker in woods near the B7015 in West Calder on December 17, with the bones later identified on Christmas Eve.
It is thought he was killed shortly after he was last seen in Glasgow eight months previously. Police are still investigating his murder.