Defense Secretary Caspar W. Weinberger's search, under White House pressure, for ways to trim the Pentagon budget has been complicated by a new internal estimate that the Navy's shipbuilding program will cost "at least" $5 billion more than anticipated over the next five years, defense officials said yesterday.

Navy Secretary John F. Lehman Jr. confirmed last night that the estimate had been made but said the Navy does not believe it. "We stand on our record of the last two years," he said. "We will come in under budget."

Weinberger sided with the Navy in a confrontation with Pentagon cost analysts on New Year's Eve, Lehman said, and kept the service full speed ahead on shipbuilding for the next five years.

However, the disputed estimate further clouds the future of Reagan's defense buildup when Pentagon budget chiefs are conducting frantic paper exercises to assess the consequences of trimming defense to help reduce projected federal budget deficits of more than $200 billion in fiscal 1984 and beyond.

Weinberger agreed when pressed at a White House meeting on Tuesday to order studies to see if cuts could be made in the fiscal 1984 defense budget without undercutting military readiness or weapons production. The asessments continued at a furious pace yesterday.

The Pentagon comptroller's office last night assessed the impact of a series of cuts, officials said, including reductions of $8 billion, $11 billion and $20 billion in budget authority and only a modest 3 percent increase in actual spending from fiscal 1983 to 1984.

Weinberger has been shuttling back and forth between the Pentagon and the White House as the Pentagon comptroller's office made assessments in conjunction with defense specialists at the White House Office of Management and Budget. The military services had not yet been asked to recommend specific cuts.

"Nothing has been decided yet," one official said. "The first round was a horror story about how much force structure would have to be reduced to absorb those big cuts. Now they're talking about much smaller cuts."

Force structure translates to Army divisions, Navy ships at sea, Air Force wings and Marine amphibious units. The Reagan administration has said it would make more sense to shrink forces by reducing personnel on the payroll than to reduce readiness by skimping on spare parts, ammunition and training.

Savings made by reducing personnel also can be achieved much more quickly than scrapping super-weapons like the Air Force's B1 bomber or the Navy's two Nimitz-class aircraft carriers, which are paid for in installments over five years or more.

The Pentagon also has estimated that saving $1 in actual spending requires cutting $4 from the budget authority sought from Congress each year. It would take a $20 billion cut in budget authority for buying weapons to reduce actual spending next year by $5 billion.

The rising costs of building ships aggravates not only this year's money problems but those for the next several as well. The Navy's projected cost overrun of "at least" $5 billion involves most of the big ships it wants in its 600-ship fleet by 1989, officials said.

Reagan and Weinberger have approved the Navy's plan to build a fleet consisting of 15 battle groups of aircraft carrier and escorting cruisers and destroyers, 100 attack submarines, amphibious ships for Marines, and supporting repair ships and oilers.

The shipbuilding plan presented to Congress last year called for building 21 new ships and modernizing two others, including a battleship, in the coming fiscal year at a cost of $12.5 billion. The five-year shipbuilding program, which would add 24 ships in fiscal 1985, 32 in fiscal 1986 and 38 in fiscal 1987, would cost about $100 billion, if last year's projections prove closer to the mark.

Weinberger and Lehman persuaded Congress last year to set aside $6.8 billion to build two more Nimitz-class nuclear-powered carriers, contending that building the two carriers together at the Newport News yard would save $750 million.