Congress began cutting spending yesterday to meet deficit targets in the Gramm-Rudman-Hollings budget law while preparing for a showdown over empowering the president's Office of Management and Budget to make the cutbacks if Congress fails to do so.

The spending surgery was begun by the Senate Finance Committee, which together with the House Ways and Means Committee is responsible for $4.3 billion of $30 billion in deficit reductions. The reductions are projected in Congress' fiscal 1987 budget to meet the $144 billion deficit target for next year.

The Finance Committee approved nearly $3 billion in Medicare cuts as part of a package that will encompass tax increases. It is expected to include an extension of the Medicare payroll tax to state and local government workers not already covered and a continuation of the federal telephone excise tax. The mix of spending cuts and tax increases apparently will be tilted more toward spending cuts than the budget resolution proposed.

In a spending increase sanctioned by the budget, the Senate panel approved a "cap" of $520 on patients' first-day hospital payments under Medicare. The charge, now $495, had been scheduled to rise to $572.

The House panel is expected to consider many of the same spending proposals today, along with possible catastrophic-illness insurance protection for Medicare patients.

The deficit reductions under consideration by the two committees, totaling $12.6 billion over three years, are to be combined with similar measures from other committees into "reconciliation" legislation that Congress will consider later this summer. Savings from appropriations bills, interest costs and other items are intended to reduce the deficit to $142.6 billion, well within next year's target.

But slow economic growth is adding to the deficit, and many lawmakers fear that Congress alone will not cut it enough, especially now that the Supreme Court has knocked out Gramm-Rudman-Hollings provisions for imposing automatic across-the-board spending cuts if the deficit target is exceeded by $10 billion or more.

To meet the high court's objections, Sens. Phil Gramm (R-Tex.), Warren B. Rudman (R-N.H.) and Ernest F. Hollings (D-S.C.) are proposing to empower OMB to implement the cutbacks, thereby circumventing separation-of-powers flaws in the original law. They plan to offer the proposal as an amendment to legislation that Congress must pass within the next couple of weeks to extend the federal debt ceiling.

But some senators say the proposal would give too much power to OMB, and the House Democratic leadership is cool to the idea. Moreover, the debt measure could become mired in controversy over a host of other issues.

OMB Director James C. Miller III yesterday endorsed the new Gramm-Rudman-Hollings proposal but warned that OMB would have to have discretion in administering the law if it is to be constitutional. He dismissed suggestions that he might abuse the powers, saying, "It'll be such a public undertaking, with so much public debate around it, that a person would really have to be crazy to do it abuse the law's intent ."

The Finance Committee, in dealing with Medicare, made cuts and improvements to arrive at its net reduction of $3 billion.

The committee agreed to limit the increase in the rate paid to hospitals under Medicare to 1.5 percent for fiscal 1987, less than the hospital rate of inflation but more than the 0.5 percent boost proposed by the administration. The panel also voted to make employers provide their disabled workers insurance similar to that given other employes, with Medicare to become the secondary payer to the private policies. Action on a major change in paying hospitals for capital outlays was postponed until today.

The panel voted to wipe out a program under which the government makes Medicare payments for hospital costs every two weeks and instead would allow 24 days -- a device that would "save" Medicare more than $2 billion in 1987 by delaying some payments past the end of the year, but ultimately would produce no real savings.

In Medicaid improvements, the committee would allow states to expand coverage for low-income children and certain blind and disabled adults whose incomes are higher than the state Medicaid cutoff, but below the poverty line.