The 70,000-dwt Formosa Falcon (built 2005) lodged off Grand Bahama on 26 September.
The latest bid to shift it came at high tide on Saturday, the islands' transport ministry said.
A tug is on stand-by next to the Taiwanese ship.
A dive inspection of the tanker revealed that the hull was not breached.
Containment booms have been deployed as a precautionary safety measure.
The plan now is to remove some of the 13,000 tonnes of cargo from the ship to lighten it.
Local
