Officers cordoned off the A198 close to Gosford House in Longniddry after the grim discovery on Sunday evening.
Detectives said it was too early to say whether the remains might be those of Louise Tiffney, 43, a single mother, who vanished from her Dean Village home in May 2002.
Police investigating her disappearance believed she may have been buried on or near the 60,000 acre estate surrounding the house and members of her family remain convinced this is her likely final resting place.
“There are a number of considerations upon the discovery of human remains,” said Detective Superintendent Pat Campbell. “At this early stage of investigation while we’re still working to establish the identity of the remains it would be inappropriate for us to speculate further.
“If any persons become relevant efforts will be made to contact their family as soon as possible.”
A member of the public called the emergency services when they made the discovery at about 6.30pm on Sunday.
A wedding and corporate venue, Gosford House was used in the filming of hit fantasy TV series Outlander, starring Sam Heughan and Caitriona Balfe.
Police are now scouring the site for clues as to the identity of the remains and the circumstances around the death, which is currently being treated as “unexplained”.
Det Supt Campbell added: “The recovery will be a painstaking process.
“We are in the very early stages of this investigation, to recover the remains, identify the deceased and inform their family, and establish the circumstances of the death.
“I would not wish to cause any unnecessary distress to families or anyone who is waiting for news of a missing loved one, and my officers will be in contact with the next of kin of any person who becomes relevant to the investigation.
“The area where the remains have been found, including the A198, will remain cordoned off whilst we conduct our inquiries and I would like to thank the public for their patience during this time.”
Mystery surrounds the disappearance of Ms Tiffney, who worked at the Raeburn Laundrette in Stockbridge and had a nine-year-old daughter, Hannah.
Louise’s son, Sean Flynn, then 21, was tried for her murder in 2005 but walked free after a verdict of not proven.
The Crown had alleged he killed his mum after a row over his conviction in June 2002 for causing the deaths of two friends in a car crash.
A trial heard he was the last to see his mum alive and blood stains were found in his car boot at the time of her disappearance.
After being cleared of murder, Mr Flynn is thought to have left Scotland to start a new life.
Louise’s sister, June Tiffney, begged police after the trial to search the area around Gosford House – convinced her sister was buried there. Detectives are understood to have combed areas of the estate but ruled out a complete search because the 60,000-acre estate was too big.
Undeterred, June and Louise’s other sister, Iris McKinlay, appealed for the help of divers and outdoor specialists to launch their own search but her body was never found.
The bones are understood to have been discovered on land outside the boundary wall of the estate, close to the shore and the A198 coast road. The discovery is thought to have been made on land between the beach and nearby woodland where the conditions are likely to be good for preserving DNA evidence.
Gosford Bay is also linked to the World’s End murders as serial killer Angus Sinclair left the body of one of his victims, 17-year-old Christine Eadie, there.
Christine’s naked body was discovered by hill walkers the day after the pair vanished in 1977. The body of her friend, Helen Scott, was found unclothed six miles away in an East Lothian field. No attempt had been made to conceal their bodies which had been bound and gagged.
A former senior detective in Edinburgh told the Evening News police would be starting a thorough fingertip search around where the body was found at Gosford.
“The first thing to do when they find remains is to treat the area like a crime scene.
“It could be an old burial site or a missing person but they’ll cordon off the whole area to make sure it’s secure. It’ll be treated as a crime scene with the forensics people down there taking soil samples.
“CID will be involved doing a really, really, thorough hands and knees search of that area. Then they’ll take the remains to the pathologists to find out how old they are because sometimes bones can survive for hundreds of years depending on soil conditions. Once they get a rough time, then they start looking at missing persons in the area and the age and sex.
“DNA can survive for a long, long time depending on soil conditions. In this area I’d expect it to be a mix of typical red East Lothian soil and sandy conditions, which’ll be pretty good.”