The Reagan administration has formally notified Congress that it now intends to propose further large cuts in Social Security, partly to maintain the giant system's solvency and partly to reduce its own future budget deficits.

The new cuts, according to a letter from Budget Director David A. Stockman and Treasury Secretary Donald T. Regan to Senate Minority Leader Robert C. Byrd (D-W.Va.) on behalf of the president, could amount to $8 billion a year initially.

The letter did not specify what form the cuts might take. But it was the first time that the administration has indicated the size of the cuts it is contemplating.

In its first round of budget proposals two months ago, the administration said basic Social Security benefits for the elderly were part of a "safety net" of programs to protect the needy that it would not touch.

Since then, however, Stockman and others have increasingly been signaling that Social Security would not remain immune forever, and two days ago Regan said explicity that the safety-net promise was for one year only.

Then last night, administration aides made available the letter to Byrd, also sent two days ago, saying the president will soon propose "reforms" in Social Security that would "reduce current-law outlays" by about $8 billion the first year.

The administration has been moved to act on Social Security for two main reasons.

First, the fund continues to be in great trouble, with benefits threatening to outstrip revenues. Democrats in the House have already begun work on a bill to correct this and have called on the administration to make its own proposals. The Department of Health and Human Services has been working to come up with some.

Second, and perhaps more important, the administration is still looking for ways to balance the federal budget in the next several years. The spending cuts it has already proposed will not do this since it also wants to cut taxes and increase the defense budget.

President Reagan's entire economic program is based on reassuring investors that in the future inflation will decline and the economy will be ripe for growth.

To achieve this, which they acknowledge they have not done so far, Reagan's advisers feel they must have a budget-balancing plan, and they have already cut in most of the easier places. That is why they are now turning to Social Security, which, at $156 billion a year, more than a fifth of the budget, is an inviting target.

Stockman is also looking afresh at the defense budgets for future years; he also is contemplating some $5 billion in cuts elsewhere in the remaining five months of this fiscal year.

This desire to balance future budgets also led the Republican-controlled Senate Budget Committee several weeks ago to propose a new system of calculating the annual cost-of-living increases in Social Security benefits (and some welfare payments an federal and military retirement benefits as well).

The committee's proposal to do this is on the Senate floor. Currently, Social Security and the other indexed benefits rise the same as prices every year, even when prices rise more than average wages in the economy. Under the committee proposal, they would rise the same as prices or wages, whichever rose less. The effect of this and a three-month delay in payment dates would be to cut cost-of-living increases for the affected programs by $7.9 billion in 1982 and similar amounts annually over the next few years.

Sen. Donald W. Riegle Jr. (D-Mich.) is trying to block the committee proposal. It is his effort that prompted Byrd to ask the White House its position and Stockman and Regan to reply.

They said the president does not favor altering the cost-of-living provisions in the current law but that the administration will shortly send up other Social Security benefit reductions "sufficient to achieve" cuts of the same magnitude.

There are many ways to cut Social Security benefits and most are under study at Health and Human Services and the White House. Most would not cut benefits for people already on the rolls -- one American in seven now receives help through Social Security -- but would mean a much barer cupboard for people going on the rolls in the future.

Among possibilities: reduce benefits more than now for people who want to retire early, before age 65; reduce the basic benefit calculation for future retirees so that the average worker's first benefit check is about 37 or 38 percent of his last paycheck instead of 42 percent as now; reduce the benefits for widows, children and nonworking wives going on the rolls in the future (for example, cutting the wife's benefit from 50 percent of her husband's to perhaps 40 percent); reduce the extra benefits for low-income workers; require coverage for newly hired federal, state and local government employees; make some adjustment in the cost-of-living increase, even though the president seems to have ruled this out for now.