The leader of a trio of Republican senators who sidetracked President Reagan's budget in the GOP-controlled Senate this month said yesterday that an agreement is near to resolve the dispute and get the budget moving again.

"We want to support [the president], and I'm confident we're going to get the details worked out very soon," Sen. William L. Armstrong (R-Colo.) said in an interview on NBC-TV's "Today" show, confirming reports from other Senate quarters that Republicans might have found a facesaving way out of their embarrasing snarl.

Final details will not be worked out until a caucus of Republicans on the Senate Budget Committee Monday when Congress reconvenes after a recess that began April 10. If agreement is reached, the committee may approve a budget plan Tuesday.

Reagan received his biggest Capitol Hill setback on April 9 when Armstrong and Republican Sens. Charles E. Grassley (Iowa) and Steven D. Symms (Idaho) joined Democrats on the Budget Committee in voting against a committee-drafted budget resolution tailor-made for Reagan's economic program.

The Republicans demanded that more savings be achieved so the budget can be balanced in 1984, as Reagan has promised. "That isn't easy but I'm convinced it can be done, and frankly I have very little doubt that on Monday or Tuesday that we'll do exactly that," Armstrong said in the interview.

Under general outlines worked out so far, a Budget Committee source said, the projected deficit for future years, including the $44.7 billion deficit estimate for 1984, will be reduced through a combination of targeted spending cuts and "language saying that more savings will be made."

Progress toward resolving the dispute was achieved in a series of meetings Thursday involving the three Republican holdouts, Senate Majority Leader Howard H. Baker Jr. (R-Tenn.), Senate Budget Committee Chairman Pete V. Domenici (R-N.M.) and White House officials.

"The question is how do we spell out the details, how do we put it on paper," explained Armstrong. "We haven't quite achieved that yet, but we're going to get it done."

The Senate already has approved most of Reagan's proposed spending cuts for fiscal 1982 in the form of program-reduction instructions to its legislative committees.

The budget resolution proved more controversial, however, because it included revenues and deficits as well as spending. Democrats' objections will not matter if Republican defections can be avoided. Republicans control 12 of the 22 seats on the committee.