Tazarvifard Mohammed Ali, 31, was last seen at the hospital in Salamanca Street, Parkhead, about 1pm on Friday afternoon.
Mr Ali, who is Iranian, is described as dark-skinned, 5ft 10in tall, of slim build with short dark hair and a short trimmed black beard.
He was last seen wearing a black puffer jacket, dark jeans and grey Nike trainers.
Inspector Cat Sutherland, Shettleston Police Office, said: “Mr Ali, who speaks good English, had been receiving treatment at the hospital recently, however, had left around 1pm but with a view to coming back at some point later in the day.
“When he hadn’t come back around 4.30pm, staff called police.
“He needs to take daily medication which we don’t believe he has with him, and as a result of that and some recent personal issues, police are concerned for his well-being.”
Police are checking CCTV at the hospital and searching the area around his home in Strowan Street, Sandyhills, Glasgow, but so far there have been no further sightings of him.
One line of inquiry police are pursing is that around 4.05pm on Friday afternoon, a man was seen to fall into the River Clyde near to Clyde Street, Glasgow.
He is described as in his 40s, of Middle Eastern appearance, with sallow skin, short black hair, black greying stubble and wearing a thick black jacket with a square on the shoulder that had a yellow border.
The inspector added: “At this time, we are not saying that this was Mr Ali, it may very well have been someone else, but we cannot rule out the possibility that it might be him.
“Officers, supported by the force helicopter and colleagues from HMS Coastguard and Scottish Fire and Rescue Service, have been and are today searching the area where that man was last seen, however, he has not yet been seen or traced.”
Anyone with information is asked to call Police Scotland on 101.