President Reagan's hopes for Republican unity on behalf of his latest budget plan fizzled yesterday in Congress as House and Senate GOP leaders proclaimed agreement and then instantly marched off in different directions.

At best, the leaders could offer the prospect of only modest savings for fiscal 1982 as they set a general goal "in the range of $115 billion" in deficit-reducing steps over the next three years, with no guarantee that even this would achieve Reagan's objective of a balanced budget by 1984.

The leaders plan to meet with Reagan early next week before deciding on firmer budget-savings targets.

One of the main points at issue is whether Congress should commit itself this year to enough tax increases and cuts in basic benefit or entitlement programs to bring Reagan close to the $16 billion in further savings that he wants for this fiscal year.

Senate leaders want to do so, according to a Senate GOP leadership aide. But House Minority Leader Robert H. Michel (R-Ill.) said his colleagues, as a minority in the Democratic-controlled House, could make no such agreement. Any tax increases "would open a can of worms over here," said Michel, and additional entitlement cuts are also doubtful.

Without action on revenues and entitlements, Reagan would have to settle for whatever he could get out of appropriations bills, which is likely to be $5 billion to $6 billion, according to Senate Appropriations Committee Chairman Mark O. Hatfield (R-Ore.) and Rep. Silvio O. Conte (Mass.), ranking Republican on House Appropriations. Reagan wants $10.4 billion from appropriations.

Hatfield's committee resumed its work yesterday and cut an additional $432 million from the Treasury-Post Office money bill, bringing it to $9.4 billion, still $252.3 million over Reagan's new targets. The committee also approved agriculture appropriations of $22.8 billion or $521.7 million more than Reagan wants, and environment and public works appropriations of $12.4 billion, or $326 million more than Reagan's recommendation.

Senate Republicans are also looking to savings of about $2.8 billion from defense, or $800 million more than Reagan proposed, according to Senate sources. But House Republicans said they didn't agree to that either.

Putting the best gloss possible on their negotiations, Michel and Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) carefully avoided specifics in announcing what Baker called "a tentative plan" to carry out the three-year, $115 billion set of new budget cuts that Reagan proposed last month to keep stubbornly rising projected deficits under greater control.

"There's only so much we can do in 1982," said Baker, choosing instead what has become a gradually evolving Republican theme of stressing three-year goals over one-year targets.

And even Baker acknowledged that $115 billion in spending cuts and new revenues "may not be enough" to achieve Reagan's goal of a balanced budget by the end of his term.

Although Baker didn't mention it, other sources said that in addition to whatever appropriations cuts Hatfield can make, the Senate strategy calls for so-called "reconciliation" instructions in the upcoming final budget resolution for fiscal 1982 to require revenue increases and entitlement cuts some time after the new year.

There has been serious talk among Senate Republicans of up to $8 billion in tax increases for fiscal 1982 that could yield $50 billion or more over three years, although Baker indicated yesterday that an increase in excise taxes on tobacco and alcohol, which has been mentioned as an option, is unlikely for this year.

Talk of sizable tax increases, coming as the country enters a recession and Congress enters an election year, has not gone over well in the House, all of whose seats are up for election next year. In a session late yesterday with reporters, Michel all but ruled out any advance commitment to "revenue enhancement," as the administration describes selective tax increases, and appeared cool to the idea of moving ahead now on entitlement cuts.

House Democratic leaders also take a dim view of another bout with "reconciliation."

"By bitter experience, you get burned," said House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.).