Mr Cunningham worked for the council and his body, dressed in a smart suit, was found early yesterday morning in an ornamental garden on the roof of the council’s headquarters.
Police insist there are no suspicious circumstances around the death of Mr Cunningham, who was believed to be in his 30s and worked as a finance officer.
Council workers who saw his body at Waverley Court claim he had an injury to the back of his head.
The roof of the administrative headquarters in East Market Street is on several levels and one theory being considered is that the man fell from a higher floor.
It is not clear whether he died after work on Monday or on Tuesday morning just after turning up for work. It is understood his body was spotted by a security guard shortly before 8am on Tuesday.
On Tuesday, staff were asked to lower window blinds wherever possible to prevent workers from seeing Mr Cunningham’s body slumped on a section of the roof one storey above street level.
One worker, who asked not to be named, said: “I worked in the same office as him and we are all really shocked as he was so young.
“There were lots of police everywhere when I went in to work. I was told by a security guard that he had an injury to the back of the head.”
Another worker, who also asked not to be named, said: “It’s an area where workers can have a coffee and catch some sun on nice days in the summer.
“A lot of people had an awful fright. I mean, imagine turning up for work like normal and being confronted with a body. It’s like something out of a detective show or a Rebus novel.”
A Timothy Cunningham is listed on Linked In as a finance officer in the research and development section at the council.
Pictures of the scene show what appears to be a white police tent covering part of the rear roof garden, just yards from an ornamental cow.
Lothian and Borders Police confirmed they were not treating the death as “suspicious”.
The building, where almost 1800 staff work, is next to Waverley Station in Edinburgh city centre.
The building was officially opened by the Princes Royal in 2007.