Mark Conway, 26, was discovered in a house on Wednesday morning less than an hour after he was seen walking several streets away.
The victim, from Thornliebank in East Renfrewshire, and who was known as 'Keddy' to friends, died at the scene.
Detective Chief Inspector Jim Downie, from Strathclyde Police, said: "We know so far that Mr Conway was walking on Skirsa Street in the Cadder area around 7.25am and a short time later he was found seriously injured within the flat in Scaraway Street.
"However, at this stage we do not know exactly where or how the young man came about his injuries.
"Officers are carrying out door-to-door enquiries and scouring local CCTV footage to try to piece together his movements prior to being found."
Police appealed to anyone who was aware of a disturbance in the area to get in touch immediately.
They also asked for information on any possible sightings of Mr Conway.
He is described as 5ft 9ins with short dark hair. He had a thin build and face and was wearing a dark blue North Face jacket. He may have had the hood up.