The Navy is considering sinking its old radioactive submarines in the depths of the ocean, fresh evidence that the arms race is compounding the gigantic problem of disposing of nuclear wastes.

The Navy has five old nuclear-powered submarines waiting for burial, either in the ocean or on land. More subs will soon be lined up behind them.

"With over 100 nuclear-powered submarines in operation, the Navy is faced with eventual decommissioning of these ships at a future rate of possibly three or four per year over the next 30 years, and a permanent means of disposal must be developed that is environmentally acceptable," the Navy said in a statement signaling its intention to start assessing burial sites.

Today's nuclear attack submarines cost $850 million each, while the giant Trident missile sub has a price tag of about $1.3 billion.

One option for disposing of worn-out submarines is to take the nuclear fuel out and then dump the ships in deep parts of the Atlantic or Pacific. The other is to cut out the radioactive sections housing the power plant and send them to the Energy Department's burial grounds in Savannah, Ga., and Hanford, Wash.

A Navy spokesman said yesterday that his service is taking the first step toward finding a final resting place for nuclear submarines: preparing an environmental impact statement, a process expected to take about 18 months.

Although the Navy defangs its retired nuclear submarines by taking out the nuclear fuel, the metal in the power plant that surrounded it stays radioactive for years, with cobalt 60 the main source of radiation. However, the Navy stressed in its formal notice--printed Jan. 14 in the Federal Register--that it would take great pains to protect the environment and did not believe radioactivity from the subs would turn out to be a high risk.

Stressing that it has not decided which way to go, the Navy said sinking the submarines in deep water would be cheaper than land disposal. "If the sea disposal option were to be selected, the submarine reactor plant would be defueled and all the nuclear fuel would be removed from the ship," the Navy said. "Hull integrity would be restored and the ship prepared for towing and for flooding in such a matter that it would land on the ocean bottom intact with reactor plant containment maintained."

The Navy conceded that the radioactive metal in the reactor would rust out eventually, no matter whether the subs were buried in the ocean or on land, but it estimated that the release of radiation would be at safe, "negligible" levels by then. The study under way is designed to assess the risks.

If the Navy opts for sea disposal, the Environmental Protection Agency would have to grant a permit for a specific part of the ocean--a process that could take up to three years, including time for possible consultations with foreign governments as required by the London Dumping Convention. Land disposal would be quicker, with the administrative work taking between one and two years.

The Navy has been investigating as potential dumping grounds an area of the Atlantic 17,000 feet deep, 200 miles southeast of Cape Hatteras, N.C., and a spot in the Pacific about 14,000 feet deep around 150 miles southwest of Cape Mendocino, Calif.

Two U.S. nuclear submarines lie at the bottom of the Atlantic--the Thresher, lost in 1963, and the Scorpion, which sank in 1968. No one survived either sinking. There is also the likelihood that some Soviet nuclear-powered submarines have sunk. The Soviet Golf-class submarine that the Glomar Explorer tried to raise off Hawaii in 1974 was diesel-powered but carried nuclear missiles. It sank in 1968.

The Navy said it has detected radioactivity from cobalt 60 in the sea bottom near the Thresher and the Scorpion but not in the surrounding water, marine life or debris.

The Navy in 1959 dumped part of the power plant of the submarine Seawolf in 9,000 feet of water 120 miles off the Atlantic coast. "No significant effect on the marine environment is expected," the Navy said.

Concern about the health risks of burying radioactive material at sea or on land is sure to mount once the Navy tries to win public acceptance of a specific plan. Now awaiting decisions on their final resting places are the attack submarines Nautilus, expected to stay on land as a monument, Triton and Halibut and the missile subs Theodore Roosevelt and Abraham Lincoln.