Amazon.com Inc. is scrambling to figure out how to use drones to deliver packages over land. Another giant in the shipping business is trying to do the same thing at sea.

A.P. Moller Maersk A/S, the world's largest containership operator by capacity, is studying deploying drones aboard its giant vessels and at its port operations around the world in an effort to cut the cost of supplying ships at sea. While studies are preliminary, Maersk says it could save up to $9,000 per ship in annual operating costs by shuttling everything from mail, medicine and spare parts by drone. The company is looking at stationing drones aboard ship for other tasks, like hull inspections.

Modern vessels--many of them too big to tie up to a pier--rely on a fleet of auxiliary ships and barges to supply them with everything from fuel to food, even when they are in port. Drones could provide an easier way to move smaller cargoes quickly and more flexibly from shore to ship and the other way around.

"We normally try to consolidate our deliveries by barge to cut costs. With the drones, smaller deliveries can be more frequent, depending on the urgency," said Tommy Thomassen, technical director at Maersk Tankers, one of Maersk Group's three main units.

In January, it test flew a drone from a tugboat to a tanker off the Danish island of Zealand, delivering a small package--a box of Maersk-branded Danish butter cookies, weighing about 2.6 pounds. The cookies were dropped from a height of about five meters and didn't break. The test drone, which was supposed to cover a distance of about a mile, flew about 270 yards because of fog.

A system that could carry packages up to 20 kilograms, or about 44 pounds, "opens up all kinds of possibilities for deliveries including spare parts and other supplies," Mr. Thomassen said.

Maersk also thinks longer-range journeys could one day be a way to deliver parts to a ship with engine trouble, or medicine for a stricken crew member in the open ocean. Currently, helicopters and airdrops from longer-range aircraft are often the only option for emergencies at sea.

Maersk is also considering whether carrying drones aboard ship--including to make periodic hull inspections, for instance--makes sense. Vessels may just carry drones on board to determine on the spot how best to use them.

The move is just one more attempt by businesses to find commercial uses for increasingly sophisticated drones. Amazon.com, the online shopping giant, has started to gradually move into the shipping business itself. The company this month teamed up with Air Transport Services Group Inc. to run a fleet of 20 Boeing Co. 767 freight planes, to reduce reliance on carriers such as United Parcel Service Inc. and FedEx Corp. Its Amazon Prime Air has been on a hiring spree in the U.S., U.K. and Israel, with the aim of using drones to deliver packages short distances by air.

Apart from Maersk, the maritime industry has been slowing embracing drones. The European Maritime Safety Agency and the European Space Agency hope to use drones as early as this year to enforce new pollution standards in some of Europe's busiest waters.

In Korea, the Busan Port Authority has said it would use drones to better police ships that illegally anchor in sea lanes near the port. And in Japan, the government has plans for a drone-based maritime monitoring system that would help it track ships in Japanese waters, survey weather and respond to emergencies at sea.

Regulators have scrambled to keep pace with drone development on land. Drone use at sea wouldn't necessarily raise some of the same concerns about noise, privacy and safety that they have in many other jurisdictions. But international laws that govern conduct at sea outside of territorial waters would likely have to accommodate drones if they became widely used.

Write to Costas Paris at costas.paris@wsj.com and Robert Wall at robert.wall@wsj.com