Ambulance Service staff alerted police after the 37-year-old man was discovered on Leamington Terrace, near Bruntsfield Place, at about 05:35.
The death is being treated as unexplained and detectives were particularly keen to trace a couple who may have tried to help the man earlier.
They have appealed for anyone with information to come forward.
Det Insp Alan O'Brien said: "Our inquiry is at an early stage and we are currently working to establish the man's identity.
"However, I encourage anyone who may have noticed or spoken to this man and can provide potentially vital information on his movements to please contact police immediately.
"In particular, we understand that a couple had attempted to help the deceased male and I appeal to them to come forward as soon as possible."
An investigation is under way after a man's body was found lying on a street in the Bruntsfield area of Edinburgh.