President Reagan will ask Congress next month to more than double the Navy's shipbuilding budget from $8.8 billion this year to just under $19 billion in fiscal 1983, sources said yesterday.

This big jump, part of Reagan's record-high peacetime defense budget, is expected to be hotly debated because of the growing vulnerability of American warships to Soviet nuclear weapons and the cost of building a 600-ship fleet.

Critics for years have accused the Navy of planning its fleet on the shaky assumption that any future war would be fought with conventional, not nuclear, weapons. But the Navy has recently acknowledged that it must change its thinking in this regard.

Rear Adm. Powell F. Carter, director of the Navy's strategic and theater nuclear warfare division, said in testimony just released by the Senate Armed Services Committee, "We have revamped the training command in order to teach our people to think nuclear."

The admiral, in secret hearings by the Senate Armed Services strategic subcommittee, said today's Navy is ill-equipped to defend ships against nuclear weapons fired by Soviet aircraft, surface ships or submarines.

"The Navy's ability to survive, fight and prevail in a nuclear conflict is continuing to degrade," Carter said. "I think it is probably at its lowest ebb right now than it has been since the early 1950s."

The admiral said the Navy plans to build a better missile than it has to stop Soviet warheads headed toward Navy ships in wartime. "We are working hard on improving ship survivability," Carter told the subcommittee.

Former defense secretary Harold Brown said in an interview shortly before he left office last year that his biggest worry was how to keep American surface ships from being sunk by nuclear-tipped cruise missiles.

If lawmakers, in reviewing Reagan's shipbuilding budget, conclude the fleet is vulnerable to Soviet nuclear attack, they are expected to intensify demands that the Navy build a larger number of ships with the same amount of money so there would be a greater chance of some surviving in wartime.

Reagan's new shipbuilding proposal, however, does not point in that direction. Instead, the president is proposing to build two nuclear aircraft carriers costing $3.5 billion each; two Trident missile submarines for $1.2 billion each, and three Aegis CG47 cruisers for $1 billion each.

In addition, Reagan's shipbuilding plan for fiscal '83 calls for construction of two Los Angeles class nuclear submarines, an LSD41 landing ship for the Marine Corps, two FFG7 light destroyers, a hospital ship and modernization of a World War II battleship and overhaul of a carrier. Mine laying and cargo ships are also included in the new budget.

The administration's pronounced goal is to build a fleet of 600 ships and submarines to achieve superiority at sea. But several lawmakers, including Democratic Sens. John C. Stennis of Mississippi, Sam Nunn of Georgia and Gary Hart of Colorado, have questioned whether the United States will ever have enough money to build such a fleet.

Stennis has warned that annual shipbuilding budgets will soon rise above $20 billion a year if the Reagan plan is followed. Nunn has asserted there is no way, given the amount of tax dollars available, that the nation can afford the 600-ship fleet Reagan has in mind.

Hart has called for construction of smaller and cheaper ships, partly to reduce vulnerability and partly, by having more ships to deploy, to cover more of the world's likely trouble spots.

The Navy has developed some weapons for stopping incoming nuclear warheads before they sink ships. But leaders are not sure these weapons would do the job and thus are pursuing new ones.

One of today's problems in trying to defend ships is determining whether the weapon hurtling toward the fleet, as seen on radar screens, is nuclear or conventional.

Firing a nuclear warhead from the ship to stop such a weapon on the basis of a radar image might escalate a battle at sea to a nuclear one if the incoming missile turned out to be conventional.

Also, weapons specialists explain, firing nuclear weapons from a ship to stop incoming missiles risks what is called "the bloody nose effect." The explosion from the defending nuclear missile could end up hurting the crew and damaging the ship from which it was fired.