A knockdown dragout fight is brewing between mayors and governors over control of the sharply diminished federal aid President Reagan proposes to send to the cities and states, reviving a feud that was set aside at least temporarily, the last time Republicans occupied the White House.

Reagan promised the governors in February and again this month that he would cushion the blow of his domestic budget cut by fighting for increased flexibility in the use of funds.

But the president's proposals to merge almost 100 narrow categorical programs into seven block grants are hung up in congressional committees, with much of the opposition being orchestrated by mayors who say they would rather take their chances with federal officials than with their governors and legislatures.

While the White House is leaning on what allies it can find in randon city halls to soften the opposition of the mayors' Washington lobbyists to block grants, it is likely that what has been a back-alley fight will soon turn into a public brawl.

A group of mayors is coming to Washington this week for a press interview of a staff report from the U.S. Conference of Mayors, expected to be critical of states' "insensitivity" to the big cities and their problems.

At the Natinal Governors Asscoiation, staff members are passing out counterpropaganda in the form of studies showing growing state efforts to meet urban needs. These are preliminaries to the heavy blows expected to be exchanged at the annual conventions of the two groups: the mayors in Louisville in June and the governors in Atlantic City in August.

Behind the fight is the decision by Reagan's budget-cutters to reverse two decades of steady and sometimes spectacular growth in federal aid to states and cities. In constant dollars, ajusted for inflation, that growth has ranged from 5 percent to 12 percent a year for most of the past 20 years, according to Richard P. Nathan of Princeton University, a student of the subject.

By Nathan's calculation's, Reagan is proposing to cut that aid to states and cities by 5 percent this year and 17 percent next year.

During the growth years, state and local officials had some fights over how to divide the pie. But they found it more profitable to cooperate. During the years that Republicans held the White House with Richard M. Nixon and Gerald R. Ford while Democrats controlled Congress, state and local officials of both parties lobbied side by side for the passage of general revenue-sharing and its renewal.

But now that the pie is shrinking, a congressional staff member remarked, "they're fighting among themselves over the scraps."

As a former governor, Reagan has long advocated giving states greater discretion in the use of federal aid. At their February meeting, governors voted to support his budget cuts in return for the added flexibility that switching from categorical to block grants would give them, even though the average 25 percent reduction in the categorical grants was steeper than they had expected and the timetable for converting to block grants this year put a severe strain on their legislature and administrations.

But Gov. George Busbee (D) of Georgia, chairman of the National Governors Association, warned that " the cuts are totally unaccpetable if flexibility and relief from mandates do not arrive simultaneously. We cannot have the cuts today and the flexibility to adjust to them at some vague point in the future."

That is the greatest threat they face. Congress has accpeted Reagan's spending cuts in the first budget resoution for fiscal 1982, but none of the president's block grant bills had been passed, and some of them apparently lack the votes even to be reported from committee in anything resembling their original form.

In part, that is because national legislators, particularily the Democrats who sponsored programs targeted for merger, are reluctant to relinquish control over them to state officials. It is also, in part, because national organizations representing the clientele or beneficiaries of specific programs find their existence threatened by such mergers.

Last week, 63 such groups, ranging from the National Association of Community Cooperatives to the Southern Council for Low Income Elders, spoke against what they called "the potentially devastating consequences of the administration's block grant proposals.

But much of the opposition has come from the U.S. Conference of Mayors and the League of Cities, representing local elected officials.Citing what he called "a documented body of evidence that the states have not been responsive to urban needs." Gene Russell, a spokesman for the former group, said, "If we're going to abandon 50 years of growing federal-city relations, it ought to be a subject of national debate."

Fred Jordan, spokesman for the League of Cities, said that organization supported parts of Reagan's economic plan but "is strongly opposed to wholesale handling of responsibilities and programs back to the states."

In the developing struggle, the governors and the National Conference of State Legislatures and the National Association of County Officials find themselves generally supporting the administrations proposals, though individual officials dissent and the organizations see ways in which the block grant proposals could be improved.

The two groups of mayors are alike in their criticism of block grants, as they were in opposition to Reagan's budget cuts. But the League of Cities, representing a broader range of mayors and headed by a pro-Reagan Republican, Indianapolis Mayor William Hadnut III, is viewed by the White House as more amenable to persuasion.

"We are doing what we can to persuade our friends among the mayors that it's in everyone's interests to avoid a confrontational situation," said Rich Willamson, Reagan's staff assistant for intergovernmental relations.

But an official of the Governors Association said, "It's pretty clear that the city lobbyists don't think they have the votes to take Reagan on directly, so they will take on the states. And we have no choice but to defend ourselves."