The blaze broke out around 4.00pm (1130 GMT) on the fourth floor of the government-run ESIC Kamgar hospital in the northern suburb of Andheri.
Six people died, according to a statement released by the civic authority's disaster management cell. Around 150 patients, doctors and nurses were rescued.
"They were rescued from different floors by firemen using ladders," a disaster management official told the Press Trust of India news agency, adding that they were then taken to other hospitals for treatment.
Fire in India's financial hub of Mumbai kills 4, injures 16
He added that the cause of the fire was not yet known.
Accidental fires are common across India because of poor safety standards and lax enforcement of regulations. They are particularly common in densely populated Mumbai, India's financial capital.
In December last year at least 14 people were killed when a huge blaze tore through a popular restaurant in the city.
Earlier that month a fire swept through a Mumbai sweet shop, sparking a building collapse which killed 12 sleeping workers.
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