The long congressional struggle over spending and taxes this year has illuminated the deepening fissures in the Republican Party that foreshadow a battle for its future and for the mantle of President Reagan.

Although Congress passed a compromise budget resolution Thursday, the disagreements that erupted between "supply-siders" and traditionalists, and between House and Senate Republicans, are certain to surface again in September over tax revision and spending.

Moreover, the disputes have increasingly taken on the shape of a contest for the 1988 Republican presidential nomination, according to interviews with GOP political strategists, pollsters and members of Congress.

Two of the party's prospective contenders for the presidency, Senate Majority Leader Robert J. Dole (R-Kan.) and Rep. Jack Kemp (R-N.Y.) have come to symbolize the the main camps jockeying for po- sition in the GOP, while Vice President Bush, a third contender, has been all but invisible.

"We're moving beyond the Reagan era," said political scientist Norman J. Ornstein of the American Enterprise Institute, a think tank. "It's a time when the Republicans are beginning to grapple with not only the individual succession but also the future of the party."

The ideological rivalry of recent months has largely been between the "supply-siders," a Kemp-led group stressing the importance of lower tax rates, and more traditional Republicans, a Dole-led group stressing the need for reducing budget deficits through shared sacrifice in cutting popular spending programs.

These camps have been at odds since the nascent days of the Reagan presidency. One leading party member calls them the "New Testament" and the "Old Testament" Republicans. Others have described the contest as one between the "free-lunch" crowd and the "root canal" people.

But all agree that Reagan was able in his last two presidential campaigns -- and during much of his first term -- to borrow from both points of view, selling a blend to the American people. Reagan offered the promise of tax cuts in his campaigns and, once elected, demanded the sacrifice of spending cuts.

But as the Reagan years wind down, some Republican activists are predicting an internal struggle for dominance that could tear at the fabric of the GOP.

"After Reagan won the nomination in 1980 , the party was able to unite. The question now is, can the party stay united?" asked a key strategist in the last two Reagan campaigns. Another senior Reagan campaign veteran answered that the disputes can be withstood "as long as they don't get too nasty."

"By and large, our face to the public still is pretty clear. We're for lower taxes, less government spending and stronger defense. Ronald Reagan and Republicans stand for those. Reagan has been able to do what others hadn't for 30 years -- convince the public what Republicans stand for. As long as they don't get too nasty, we can live with them.

"But the other side of it is if it gets so vicious, and our coalition is fragile enough, that it knocks it apart. I think there is some threat there. I don't want to downplay it."

One source of GOP tension is the differing goals of House and Senate Republicans. In the Senate, Ornstein said, "you have majority Republicans who want to go to the country in 1986 on a record of responsible government."

But House Republicans "have been in the minority for 30 years, and if the country believes we have been governed responsibly, we will elect Democrats in the House. They are looking for leverage. They don't want to give the Democrats issues to use against them -- Social Security being the prime one.

"The House Republicans remember in 1982 they were kicked from one end of the country to the other by the Democrats' slogan, 'Save Social Security.' They lost virtually everything they gained in 1980. The Senate Republicans held their own in '82. So there is a division on that issue."

Dole made it clear last winter that he was staking his hopes on Republicans trimming the budget deficit by more than half, saying repeatedly that voters will reward Republicans if they demonstrate an ability to govern this year by successfully tackling it. The majority leader had almost constant support from then-director of the budget David A. Stockman in attempting to impose shared sacrifice on the political system -- including in Social Security benefits.

But when the White House went along, House Republicans rebelled. Kemp said this week, "My view, and that of some others, is that Social Security is not an issue around which the Republicans needed to balance the budget or confuse senior citizens on."

Kemp helped scuttle Senate Republican plans for raising taxes and restraining Social Security benefits, and found a receptive ear at the White House. He has staked his hopes on further tax-rate reductions.

It is the tax-overhaul effort this fall that may reignite the Kemp-Dole competition. Many Republicans say that they expect that an effort to raise taxes will be made in the Senate.

Bush has intentionally stayed out of the party conflict. He has staked his hopes on a successful Reagan second term and Reagan's popularity. In recent speeches, Bush has seemed to embrace both ideological camps, calling for spending cuts but resisting tax increases. "He's decided what his game is," and it is loyalty to Reagan, said an aide.

A senior administration official said the jockeying may be forgotten by 1988. "The most important thing for the Republican Party in '88 is that the economy be strong," he said. "That is more important than all these other issues. If it isn't, I think it's academic who the Republican nominee is."