Lionel Simenya, 36, from Burundi, is thought to have become involved in an "altercation".
There were reports today he was asleep in a car which joyriders crashed into.
He was found early on Thursday in Ford's Road, off Gorgie Road, in the west of the city.
Police said they were alerted around 3:50am and said he had been found to have suffered serious injuries but dismissed reports he had been knifed.
Detective Inspector Stuart Alexander from Police Scotland's major investigation team, said: “First and foremost, our deepest sympathies are with the family of Lionel Simenya and we are pursing various lines of enquiry to identify those responsible for causing his death.
“It is understood Mr Simenya was within his vehicle in Ford's Road and has become involved in an altercation.
"Although investigations are at an early stage, there is nothing to suggest that he has been a victim of a knife attack.
“I have a full team pursuing various lines of enquiries and I am particularly keen for anybody in the surrounding area who has private CCTV or dashcam footage from the early hours of Thursday morning to contact us.
“We are making enquiries to ascertain whether a Peugeot car that was stolen on Thursday morning on Ford's Road and found abandoned in the same street is connected.
“Lionel Simenya moved to the UK a number of years ago.
"He was a highly thought of, hard working man who kept himself to himself and has met a tragic death.
“I am confident the answer to solving this horrific crime lies in the communities of Edinburgh and no matter how insignificant you think any information is, please contact us and let us assess it.
“This must be playing on the consciences of the individuals responsible and I would urge those people to come forward.”
In a statement issued through Police Scotland, the family of Lionel Simenya said: “We are profoundly shocked and extremely saddened that our beloved Lionel has been taken from us in such a cruel manner.
“Lionel was a hard-working and dedicated chef, who had won an award for his skills and was being considered for employment at the Scottish Parliament.
“We would ask anyone who can help police with their investigation to get in touch and provide any information that can bring those involved in his death to justice.
“Anyone who was within that car should search their conscience and realise that our family have been left devastated by their actions. Hopefully then they will do the right thing.”