Officers found the man lying in the flat in Hawkhill Close, in Lochend, but he was “unresponsive and had passed on”.
An ambulance was called to the six-storey apartment but the body of the man, who is believed to be in his 30s, was still in the flat yesterday afternoon as detailed forensic tests continued.
Neighbours reported hearing raised voices followed by a loud bang shortly before 8pm on Tuesday night when it is believed the man was killed.
Police chiefs would not confirm that the victim had been shot and said only that they were treating the death as “suspicious”.
Investigators have ascertained the name of the occupant of the flat, but they are still working to confirm whether he is the dead man.
No identification of the victim had been made today.
A neighbour, who lives opposite the deceased on the sixth floor, described the person who lived in the flat as a man, aged between 30 and 35 years old.
He said: “He lived alone but there would be people coming and going at all hours of the day and night, from early in the morning until late.”
The development, near Hibs’ Easter Road stadium, was cordoned off while police searched bushes and undergrowth for evidence.
White-suited forensic officers could be seen on the landing of each floor and sniffer dogs were also deployed in the surrounding area.
Durza Zeri, 50, said: “I came back on Tuesday night and wasn’t allowed to enter the complex and had to park across the road. There were police officers everywhere but they wouldn’t tell us anything. It’s incredible to think something like this has happened.”
Superintendent Matt Richards said: “We received a call at 8.15pm on Tuesday and we attended as a result of local concerns. We forced entry to the address and found the man within the flat. An ambulance was called but it was clear he was unresponsive and had passed on.”
Supt Richards would not be drawn on whether police believed the man had been deliberately targeted, but added: “There is no reason to believe it will be repeated.”
He said: “We are at an early stage of our enquiries, however the man’s death is being treated as suspicious.
“Officers will be carrying out high-visibility patrols in the local area in the coming days in order to provide reassurance to local people. However, there is not believed to be any risk to the public at large.
“We are appealing to anyone who was in the area of Hawkhill Close on Tuesday evening, who saw or heard anything suspicious, to contact police as a matter of urgency. Likewise, anyone else who has any information.”