Big-city mayors told the nation's governors today that they need broader authority from the states to levy local taxes if they are going to absorb the budget cuts imposed by President Reagan and Congress.

In a panel session at the National Governors Association annual conference here, representatives of the U.S. Conference of Mayors and National League of Cities said that the combination of federal fund reductions and strict state limits on local tax sources was more than they could take.

The discussion was strained but polite, as the local officials muted their normal rhetoric and sought to keep open the door to discussion the governors opened with their invitation for the mayors to appear at this meeting for the first time in years.

Meanwhile, the governors found themselves face-to-face with their first serious internal debate in many years on the role of the federal government in education, with two leading Democrats disagreeing over whether it is time to end Washington's intervention in school finance entirely or just restrict it.

The argument that surfaced today between Florida Gov. Bob Graham and Arizona Gov. Bruce Babbitt may spark the most serious controversy on the education-aid issue since the governors accepted the need for federal financial aid to schools almost 20 years ago.

Graham, the chairman of the education subcommittee, has drafted a new policy statement that would target federal assistance to some extent but keep it as a major part of the finance mix. Babbitt wants the states to take over full responsibility for school finance as part of a major program swap that would require the federal government to pay what is now the state share of Medicaid in return.

Both are expected to argue their viewpoints at Tuesday's plenary session.

The highlight of today's discussions, which ranged over the whole field of domestic programs, was the appearance of four local elected officials before the fiscal affairs committee headed by Vermont Gov. Richard A. Snelling (R), who becomes the new chairman of the association on Tuesday.

Snelling has set as his goal for next year the "reopening of the dialogue" between state and local governments, and invited the four officials today as a prelude to that effort.

Two of them, Peoria Mayor Richard Carver (R) and J. Richard Conder (D), a Richmond County, N.C., commissioner and president of the National Association of Counties, are supporters of the Reagan program to shift many categorical federal aid programs to state block grants.

But the other two, Indianapolis Mayor William H. Hudnut III (R), president of the National League of Cities, and Nashville Mayor Richard S. Fulton (D), chairman of the advisory board of the U.S. Conference of Mayors, speak for groups that were critical of the Reagan budget cuts and argued that more of the scarce funds should be allocated directly to the cities, bypassing the states.

Both chose to temper their prepared remarks in their appearance before Snelling's committee. But they managed to make the point that the combination of federal fund cuts and state-imposed tax restrictions left the cities as Hudnut put it "on the short end of the stick."

"We cannot have more responsibility placed on us at the local level without the tax authority to do the job," Hudnut said, pointing out that in about half the states big cities lack the home rule authority to select their own tax sources and set their own tax rates.

The governors who were listening did not offer any pledges, but Snelling said that many of them recognized that the "lack of sensitivity" to city needs that Hudnut decried among state legislators was a problem "we have to deal with together."

The governors gave Environmental Protection Agency Administrator Anne M. Gorsuch a polite hearing as she strongly defended the Reagan administration's proposals for altering the Clean Air Act, a controversial plan certain to face congressional opposition.

Gorsuch described how she believed the EPA was "virtually drowning in a sea of minutiae" -- a position which meant that it was essential to loosen clean air regulations.

She said the administration's proposal for relaxing auto emission standands would have only a 2 percent overall impact on the environment -- what she called a "semi-insignificant difference" -- and she was strongly backed by Gov. James A. Rhodes (R-Ohio), who denounced the effects of the Clean Air Act on industrial states. He said of Gorsuch: "Now we have got a lady to do a man's job."

Gorsuch said the administration was concerned to maintain environmental standards but contended that "some fine-tuning of the act to save jobs in the auto industry" could be accomplished without a significant effect on standards.

Earlier, during the same session of the committee on energy and the environment, Rhodes clashed with California Gov. Edmund G. (Jerry) Brown Jr. (D) over oil drilling off the coast of California.

Brown claimed the Commerce Department was now preparing to alter regulations in an effort to get around a court decision upholding the state's right to prevent drilling in an offshore area just north of Santa Barbara.

Brown accused the department of trying to make it possible for Interior Secretary James G. Watt to allow drilling despite California's opposition, on environmental grounds, which was upheld in a U.S. District Court ruling last month.

At a session of the transportation committee, Rep. James J. Howard (D-N.J.), chairman of the House Public Works and Transportation Committee, told the governors they faced "a nation in ruins" because of the effect of the Reagan budget on public works programs.

Howard said the nation could hardly survive if major programs --on highways, seaports and water systems -- were abandoned.