The 24-year-old man was found in a bedroom in Ross House, Mount Vernon at around 11.30 am on Monday.
Police say he had sustained head and facial injuries that have left him critically ill in hospital.
Investigating officer Detective Inspector Mark McHaffie said: “At this time we believe this male may have been the victim of an attack or assault of some kind.
“We are working hard to establish exactly what happened.
“We are keen to hear from anyone with information that could assist our investigation.
“In particular anyone who was in the vicinity of Ross House from midnight to midday on Monday 28th December 2015, and who saw or heard anything that could assist.”
Anyone with information can contact detectives on the 101 number.
Alternatively information can be passed anonymously to Crimestoppers on 0800 555 111.