The man's body was discovered by a member of the public in Cadzow Street, near to Blysthwood Court, at around 1.30am.
Police cordoned off the area from the entrance of the street to the gates of Cadogan Square Car Park.
An incident tent was set up in the street below the public walkway leading from Blythswood Court's Davaar block to the other two high rise buildings in the housing complex.
The man is believed to be in his 30s.
Police said his death was being treated as suspicious.
Several uniformed officers as well as plain clothed officers were at the scene carrying out inquiries.
As well as Blythswood Court, the busy area is home to businesses.
Residents of the housing complex spoke of their shock.
A 37-year-old woman who lives in the Columbia block of Blythswood Court, said: "I didn't hear any disturbances in the night.
"When I was walking home at about 10pm I did notice a man on his mobile around the bridge area and thought something was going on but it could have been anything.
"I'm shocked to see what's going on this morning but there is often a lot of trouble around the Davaar flats.
"It seems to be getting worse."
A 26-year-old resident said: "There is a lot of trouble and violence in this area. I was attacked walking home with my messages before. But I'm totally shocked that a man has died here."
A Police Scotland spokesman said: "Police received a report of the body of a man, believed to be in his 30s, being found outside Blythswood Court at around 1.30am today.
"The area has been sealed off and police are treating the death as suspicious. Inquiries are continuing."