Officials said many of those on board the vessel escaped by jumping over the side, the Indo-Asian News Service reported.
They believe the trapped sailors may have moved to the tail section of the submarine, which is a little above the water line, CNN-IBN reported.
Sources told CNN some personnel may have died on the submarine, but the Indian Navy has yet to confirm that.
Fire officials said the blaze aboard the vessel was brought under control at about 3 a.m., nearly 3 hours after it started.
The submarine was in the Naval Dockyard, a high-security area in Mumbai.
The cause of the explosion was under investigation though sources told CNN the incident may have been caused by a gas leak.
The incident couldn't have come at a worse time as the Indian navy's submarine fleet is currently "in a highly precarious state," a secret Defense Ministry report said.
The submarine force would be at its lowest in history by 2015, the report said in April, as reported by the Hindustan Times.
The explosion has put a dent in the Navy's undersea capabilities, a top Navy official said.
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