Sharply divided for the first time about how much further it can cut domestic spending, the Republican Reagan administration is engaged in a fierce struggle over next year's budget.

Foreshadowing arguments likely to dominate congressional debate in the 1982 election year, key administration officials are resisting proposals for new spending cuts put forward by David A. Stockman, the president's budget director. On Thursday, President Reagan himself will begin to hear appeals from his Cabinet members against deep new cuts.

As he studies options for next year's spending levels, Reagan will encounter the basic budgetary dilemma:

On one hand, if he doesn't make huge new cuts in spending, he will face unprecedented annual deficits of $100 billion or more; on the other hand, all political signals indicate that Congress, the nation's mayors and governors and members of his own Cabinet oppose big new cuts.

Last spring, Reagan led a crusade to cut the budget. He exploited his landslide election victory, the newness of his administration and the disarray of his opponents and turned that crusade into a victory that surprised many. But as a ranking Senate aide put it last week, the conditions that made that victory possible have now disappeared.

The new realities for 1982 include a deep national recession and congressional elections in November.

"The difference between this year and last year," that Republican aide said, "is that this year Congress won't approve any more cuts."

Others on Capitol Hill say Reagan has somewhat better prospects. But even in the loyal Senate Republican leadership, no one is predicting that cuts can be made in 1982 as big as those achieved last summer in the "Gramm-Latta" budget reconciliation bill, which trimmed outlays by $35 billion.

Even if the administration somehow won another $35 billion in reductions, Reagan would be left with a bigger budget deficit than was faced by any of his predecessors in the White House.

This cannot be a comforting prospect for a president who declared last year: "There is but one cause of inflation, and that is government itself . . . . The cure is a balanced budget."

Signals foretelling political trouble for the administration's crusade are unmistakable. In September, the president sought legislation to reduce this year's deficit by an additional $16 billion; last week, he had to accept a compromise with Congress that will produce just $4 billion in new savings.

Members of Congress and aides in the Capitol say last year's enthusiasm for budget-cutting has vanished now that the political costs of further cuts are becoming clear.

At the White House, too, officials acknowledge that the easiest cuts already have been made, as one put it. "We're now moving into more marginal areas" where the case for further cuts is less clear, another senior official said.

Sen. Paul Laxalt (R-Nev.), Reagan's closest friend in Congress, said last week that perhaps the budget cuts made last summer had left the states with too little money to administer necessary programs. "They've got a big load dumped on them by way of new programs," Laxalt said of the states, "and we're not giving them adequate money."

Gov. Richard Snelling of Vermont, a conservative Republican who is chairman of the National Governors Association, called the Reagan economic policy "an economic Bay of Pigs."

Although Stockman has lost none of his ardor for budget-cutting and is telling Cabinet departments they must slice much deeper for fiscal 1983, this time Cabinet secretaries are resisting.

Last spring, career officials in the Office of Management and Budget commented with awe in their voices that Reagan's appointees "don't argue back," as one put it. The traditional phenomenon of departmental officials fighting for programs that benefit their constituents had all but disappeared. Now it has returned.

Commerce Secretary Malcolm Baldrige has advertised his opposition to Stockman's latest proposals, suggesting that the OMB director is following "macroeconomic theories that neither the administration nor I can subscribe to."

Health and Human Services Secretary Richard S. Schweiker also is openly fighting some suggested cuts. Anne M. Gorsuch, whom environmentalists have scourged for alleged neglect of her Environmental Protection Agency, is lobbying Republican senators for help in beating back cuts proposed by Stockman.

The Department of Housing and Urban Development has appealed cuts proposed by OMB, and Attorney General William French Smith is engaged in a behind-the-scenes fight with Stockman over new reductions.

Ardent budget-cutters inside the administration argued last week that these Cabinet members are risking Reagan's ire by resorting to open criticism or leaking their objections. They pointed to Reagan's earlier warnings that the 1983 cuts will have to be deep.

On the other side, opponents of harsh cuts inside the administration can point to substantial compromises Reagan accepted after his September proposals for additional cuts, his willingness under pressure to retain federal revenue sharing for localities, and his retreat from unpopular proposals to cut Social Security benefits.

The basic reason for opposition to further cuts is easy to find. Mayors, governors, members of Congress and Cabinet members realize that new reductions inevitably will affect popular programs, aggravate local financial crises, cut deeply into departmental responsibilities and invite strong political opposition. Cuts proposed by Stockman and already leaked show why opposition is so strong.

For example, he seeks changes that would impose major new financial obligations on localities to maintain public housing projects or force their closing. He would wipe out the government's single biggest aid program for cities.

He suggests cutting federal aid to education in half, eliminating about $8 billion, including $800 million from the "Pell grant" program of direct financial aid to poor college students and nearly $1 billion from the popular student loan program.

Stockman also would like to decimate remaining job-training programs, reduce housing benefits for the elderly and wipe out much of the Department of Energy's non-defense spending.

All of these cuts would come from straight appropriations. Stockman has not yet tipped his hand on plans for cuts in the giant "entitlement" programs that guarantee such benefits as food stamps, welfare, Medicare and Medicaid to those who qualify. These already have been cut somewhat, and further changes--requiring amendments to laws that established them and determine who is eligible--are certain to provoke sharp controversy.

There is no consensus on how far entitlement programs might be cut. A senior administration official said recently that he can imagine saving $20 billion a year from entitlements. A senior aide on the Senate Finance Committee said of that figure: "I just can't believe it."

But talk of cutting entitlements persists, not least because the administration and congressional leaders realize that the rest of the budget does not contain enough fat.

This problem is aggravated by Reagan's decisions to make no changes this year in Social Security, at $170 billion by far the biggest entitlement program, and to leave intact his ambitious defense program. Informed sources say Stockman still is hoping to persuade Reagan to trim defense.

Moreover, Reagan has rejected thus far any proposals for significantly raising tax revenues, although he is expected to adopt some relatively modest ones.

So the big entitlement programs are tempting targets, at least in principle. The White House is working in secrecy on a comprehensive plan for adjusting them downward.

Rep. Barber Conable (N.Y.), ranking Republican on the House Ways and Means Committee, warned last week that cutting entitlements "should be a more sophisticated operation" than just imposing across-the-board caps on programs. Conable noted that almost all beneficiaries of entitlements are viewed as deserving the protection of the "social safety net."

But no level of sophistication can guarantee congressional approval of potentially unpopular proposals in an election year. "You can walk it through any way you want to," said a key Senate aide on the Republican side, "approaching it from entitlements or from appropriations. The fact is, there's not much they in the White House can propose that will pass next year."

White House officials acknowledge that they will have political problems, but most economic projections show such large deficits that the administration sees no choice but to keep cutting.

OMB has set a brisk pace for the bureaucracy, which is currently preparing final appeals against Stockman's proposed cuts. In a series of six meetings beginning Thursday and ending Dec. 23, Reagan will weigh his own associates' arguments for and against new cuts.

Despite problems created for Stockman by an embarrassing magazine article last month, the budget director has kept the pressure on the departments. For example, he initially asked the Department of Energy to find $2 billion to $3 billion in savings from a $14.4 billion budget first proposed by the agency for 1983 spending. When Energy Department officials complied and produced a revised budget of $11.1 billion, Stockman promptly proposed a cut to $9.3 billion.

There have been similar exchanges for all the departments. Next, Cabinet secretaries will be able to appeal Stockman's final figures, first to a "budget review board" of Stockman, presidential counselor Edwin Meese III and chief of staff James A. Baker III, with others sitting in periodically. A few days later, the secretaries will be able to appeal personally to the president.

However sympathetically Reagan may view these appeals, he will find them hard to accept. Unless he says no, he will be unable to show in his January budget the steady decline in the deficit--from this year's expected record level--to which officials say he is committed.

Yet if Reagan backs the further cuts sought by Stockman, he could find that Congress just takes the budget decisions out of his hands.