Defense Secretary Caspar W. Weinberger has recommended to the White House that the first $20,000 of a military person's salary be made exempt from federal income taxes, Reagan administration officials said yesterday.

The proposal comes atop pay raises for military personnel that will total 26.1 percent in the 12 months ending this October if President Reagan gets his way.

Weinberger argued in heated Pentagon sessions, sources said, that addition of the tax breaks to the pay raises would help stem the exodus of skilled people from the Army, Navy, Air Force and Marine Corps.

At his insistence, defense manpower specialists analyzed tax forgiveness on the first $10,000 of income, the first $20,000, and special tax credits. The 20,000 exemption was the one forwarded to the White House Office of Management and Budget, sources said.

Administration officials gave conflicting versions last night of where the Weinberger initiative stood. One said OMB is studying it. Another said the proposal was shot down at a recent Cabinet meeting, a claim that Pentagon sources disputed.

Although Weinberger has confirmed "we're looking at" tax breaks for military people, he declined through a spokesman last night to discuss the current status of the controversial idea.

However, the fact that the new team at the Pentagon would go so far as to consider seriously tax forgiveness for service personnel suggests its growing concern as to whether the American military will have enough skilled people to operate all the new weapons systems Reagan wants to buy.

The Navy today, for example, needs 22,000 more petty officers and chiefs to man by the book the 456 ships in its fleet. And this does not count the battleships Iowa and New Jersy that the administration wants to take out of mothballs.

Looking beyond such quick fixes as reactivating battleships, the Navy's announced goal to increase today's fleet from 456 to 600 combat ships would require expanding the service by 90,000 persons.

Sen. Sam Nunn (D-Ga.) complained at a Senate Armed Services subcommittee hearing this week that the only way the Navy can get the skilled people in a hurry for those battleships and new ships on order is to entice veterans back into the service.

While endorsing the objective of a 600-ship Navy, which might include 15 aircraft carrier task forces and perhaps four battleships now in mothballs, Nunn said it was time for some "soul searching" on how the extra billions in military pay should be distributed.

Rather than continue to increase the pay of everyone by the same percentage, Nunn said, the military should scrap its "antiquated" system and pay skilled people more than unskilled ones of the same rank.

Weinberger reiterated before the House defense appropriations subcommitte yesterday that manpower is key to the administration's plan for "rearming America," and declared that "a couple of hundred thousand more personnel" will be needed, with not more than "one or two" more divisions for the land forces coming from that addition.

The idea of using tax breaks to recruit and retain high-quality military people was attacked in internal Pentagon meetings on the grounds that a $20,000 exemption would help-ranking people the most and would violate the principle that everyone must share the nation's tax burden. For example, an E-5 sergeant who is paid $910 a month would have no federal income taxes under the Weinberger proposal.

If OMB is seriously analyzing the Weinberger tax proposal, as some officials claim, it is expected to balk at the estimated $2 billion to $3 billion in tax revenue that would be lost through putting the first $20,000 of a military person's income off limits to the Internal Revenue Service.

"The tax route is not the way to go," asserted one critic last night who predicted the administration would stick with raising military pay as the way to shore up the all-volunteer force. Serviced personnel received an 11.7 percent pay raise last Oct. 1. Reagan is recommended an additional 5.3 percent, effective July 1, and 9.1 percent this Oct. 1 -- the start of fiscal 1982 -- for a total of 26.1 percent.