The box was found by UK-based Advanced Marine Services inside the post room of the SS Minden at the bottom of the Atlantic Ocean off Iceland.
There have been claims that the cargo ship was transporting gold from South American banks to Nazi Germany when it sank in September 1939.
The ship was intercepted by the Royal Navy's HMS Calypso and scuttled by its own crew so the vessel couldn't be captured by Britain just weeks after the start of World War II.
Advanced Marine Services was using the vessel Seabed Constructor to look for hold
It was claimed that the chest contains up to four tons of metal, and the British firm is now applying to Iceland's government for a permit to cut a hole in the ship's hull and remove the box.
The crew of the Seabed Constructor was previously stopped by Iceland's Coast Guard in April for not having permits to conduct research in the country's waters.
Advanced Marine Services want to bring the chest back to Britain, claiming the contents belong to the finder, the Sun reported.
Iceland's Coast Guard stopped the Seabed Constructor's crew in April
The company is awaiting a decision from Icelandic officials on who owns the wreck and its contents.
The SS Minden left Brazil on September 6, 1939 - five days after the war broke out - and there were claims that officials from Banco Germanico, a subsidiary of German Dresdner bank, helped load its cargo.
The ship was sunk by its crew nearly three weeks later on September 24.
The crew was rescued by the HMS Dunedin and taken to a British naval base.