The defence ministry, which has responsibility for the coastguard, said that an aerial survey of the slick indicated that oil had spread up to seven nautical miles from the vessel.
Coastguard and pollution control teams have been attempting to neutralise the affected area using oil spill dispersant, India's maritime watchdog, the Directorate General of Shipping, said in a statement.
The organisation said on Sunday that oil was leaking from the MV Rak Carrier at the rate of about 1.5 to 2.0 tonnes per hour.
The government of Maharashtra state, of which Mumbai is the capital, said in a statement that "there is no immediate threat to (the) Mumbai coast from this (oil)".
"There is no cause for any panic," it added.
The MV Rak Carrier was travelling from Indonesia to Gujarat state in western India when it got into difficulties in stormy weather and sent a distress call about 22 nautical miles off Mumbai last Thursday.
Its 30-strong crew was rescued before the 220-metre (722-feet) ship sank. It was carrying just over 60,000 tonnes of coal and had about 290 tonnes of furnace oil and 50 tonnes of fuel oil on board at the time.
Fishing boats have been warned to steer clear of the area, though fishing is banned until August 15 because of the annual monsoon rains.
An unmanned oil tanker, the MT Pavit, ran aground on a north Mumbai beach on July 31 after drifting undetected for more than a month across the Arabian ship.
Its crew abandoned ship after reporting flooding in the engine room and loss of power.
Copyright AFP (Agence France-Presse), 2011