The attack happened in Loch Achray Street, Sandyhills, around 9.30am on Sunday.
Emergency services attended and the 36-year-old was taken to Glasgow Royal Infirmary. He has since been released.
Police say initial inquiries have revealed the man may have been involved in a disturbance in the surrounding area which resulted in him being injured.
Officers have been gathering CCTV footage and are conducting door-to-door enquiries for further information.
Detective Constable Philip Scally said: “This area is overlooked by high-rise flats and I am appealing to local residents for any information regarding this incident to contact us.
“A man has suffered serious lacerations and stab wounds as a result of this incident so if you have any information, no matter how insignificant it may seem, please pass it on to us.
“Your information could prove vital in helping us piece together exactly what has happened.”
Anyone with information is asked to call Police Scotland on 101.