AS THE ADMINISTRATION tries to move its budget policy another step forward, a potentially formidable opposition is rapidly developing. It's manned not by Democrats--they're still out rethinking--but rather by members of the president's own party. The most vocal members of the new opposition are Republican governors and mayors who see, in practical terms, the consequences of the Reagan economic program.

Senate Republican leaders have been murmuring discontent since last summer. In recent weeks, however, much sharper notes have been sounded in public conferences and private meetings among Republican governors and mayors, many from the more conservative wing of the party. With a surprising degree of consistency they've been telling the president tht his economic program needs a change in course.

Budget cuts, these officials are saying, have been made too fast and too deep. The further cuts now being planned are insensitive to the needs of their constituents. Tax cuts were too large, too responsive to special interests and poorly designed to stimulate needed economic growth. Readjustment of past policy was needed, but present policy is an overcorrection that runs the risk of promoting a backlash in favor of a return to the liberal excesses of the past.

Thus far the administration has given the governors and mayors little more than polite attention. President Reagan says he understands that some states and localities are in dire straits, but the answer is for the discontented residents of these locations to move to places with sunnier economic climates. The president would like to help states out by turning over some federal tax sources to them, but with upwards of $100 billion deficits looming, that would be tough to do. In the meantime, his budget planners are marking up still larger cutbacks in housing, urban development, job training and other state and local aid in next year's budget.

That's not comforting news for governors and mayors in either political party. They were pretty good sports during the first round of budget cuts because they saw the real possibility of making federal money go farther if they had more say in how it's spent. They got some new flexibility, and they'll probably get more. But governors and mayors know that flexibility can't build bridges, roads and housing, feed the poor or give the unemployed the abilities they need in a rapidly changing economy. That takes money. And in an economy in which international happenings can determine the fate of local economies, tying public needs to local tax bases is increasingly anachronistic.

It won't be easy for the administration--with its repeatedly reaffirmed commitments to increasing defense spending, protecting ocial Security and lowering federal taxes--to respond to the concerns of the governors and mayors. Those officials, having found a new sense of common purpose among themselves, are not, however, likely to be shy about taking their case to their fellow politicians in Congress. The administration now finds itself in the uncomfortable position of facing a strong and vocal opposition that is growing within its own party.