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."

Attacking midfielder Christensen, 22, has played more than 50 games for the London side, but has not started a Championship match this season.
Striker Woodrow, 22, has only started one league match but scored twice in the EFL Cup against Leyton Orient.
Both players are available to face QPR on Saturday.
Brewers manager Nigel Clough said: "Lasse has a good season or so under his belt in the Championship. He's been a bit unlucky not to feature much this season, but they see him in their long-term plans and it will help him to play for us.
"Cauley is one to add to our striking options. With Luke Varney out for four to six weeks and Will Miller's hamstring injury, it was important to get another striker in."
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A suspected criminal is on the run after leaping from a hospital window and catching a bus.
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Burton Albion have signed Lasse Vigen Christensen and Cauley Woodrow on loan from Championship rivals Fulham until the end of the season.