Joseph Mensah was arrested on suspicion of assault on Sunday before being transferred to Manchester Royal Infirmary with leg injuries.
After jumping from the first-floor window, he is believed to have caught a number 50 bus on Upper Brook Street.
Greater Manchester Police have appealed for anyone with information about Mr Mensah's whereabouts to contact them.
He is described as black and of heavy build and was wearing a dark coat and jeans at the time of his escape.
Police were at the hospital at the time, but were not specifically tasked with guarding Mr Mensah.
Det Ch Insp Chris Mossop said: "We are currently investigating the circumstances that led to Mr Mensah's escape and are continuing to search for him.
"I would like to remind those who may be hiding him it is a criminal offence to do so.
"I would also make a direct appeal to Mensah as he was originally admitted to hospital for injuries to his leg and needs treatment."
A suspected criminal is on the run after leaping from a hospital window and catching a bus.