Washington ranks second among nine large American cities in the amount of private money raised by its major private arts institutions but ranks at the bottom in funding from local governments, according to a new study sponsored by the National Endowment for the Arts and the Eugene and Agnes E. Meyer Foundation.

"Although the base of philanthropy in Washington, particularly regarding corporations and foundations, is smaller than in other comparably sized cities . . . Washington's major cultural institutions can hold their own . . . " the study reported in a survey of cultural institutions in nine cities: Philadelphia, Boston, Detroit, San Francisco, Dallas, Houston, St. Louis, Baltimore and Washington.

The study is one of a number of statistic-stuffed tomes to appear in past weeks analyzing charitable giving at a time when the Reagan administration is seeking to cut federal arts funding.

Some local arts officials yesterday said they were surprised at Washington's lead in private fund-raising.

"I guess the fact that nonprofits here have survived at all in this area with the lack of local government funds is proof that we must have had" private funding, said Patricia Fleischer, director of development for The Washington Opera.

The study compared levels of private and public giving in nine cities from 1981 through 1983 by examing donations to private art museums, symphonies, operas and dance companies.

The figures for Washington represent the aggregate private and public donations to the Corcoran Gallery of Art, Wolf Trap Farm Park, Ford's Theatre, the National Symphony, the Folger Shakespeare Library, Arena Stage, the Washington Opera, the Phillips Collection, the Capital Children's Museum and the National Building Museum.

Those for Baltimore are the aggregate of donations to the Baltimore Symphony, Center Stage, the Baltimore Opera, the Baltimore Museum of Art and the Walters Art Gallery.

The study ranks Washington second only to San Francisco in total private funds raised. Washington also outranks all but Dallas and Houston in the percentage of private donations as a part of those institutions' annual operating budgets. Private donations made up 31 percent of the total budget for all 10 Washington institutions in 1983, as opposed to 24 percent for Philadelphia and 18 percent for Boston during the same period.

But the study also shows Washington trailing far behind Detroit, Baltimore, St. Louis, Philadelphia, San Francisco, Boston, Dallas and Houston in support from local governments. The percentage of nonfederal government support for Washington's private institutions was less than 1 percent of the institutions' aggregate annual budget, while the figure for other cities surveyed averaged 17 percent.

That lag for Washington was only partially offset by recent increases in federal support appropriated by Congress.

Larry Chernikoff, the study's author and current associate director of the American Association of Museums, said his conclusions do not contradict a recent study by the Greater Washington Research Center that ranked Washington last among seven cities in the number of resident major corporations as well as in corporate giving and local foundation assets and grants.

"It doesn't surprise me," agreed Tom Fichandler, executive director of Arena Stage. "We work like hell to get that money." Fichandler and others said the majority of private support comes from individual rather than corporate or foundation donations.

The D.C. Commission on the Arts and Humanities is the only source of nonfederal government funding available. Its annual budget is approximately $1.5 million.

The study suggests that federal support be increased and institutionalized to break an annual ritual in which individual Washington institutions make funding pleas directly to Congress. In an effort to streamline that process, Congress last year approved a $5 million fund for those institutions, but the Reagan administration has proposed transferring that money to the Bureau of Indian Affairs.

Officials from Washington's cultural institutions have long argued that both Congress and private corporations should foot a greater portion of maintaining the arts in the nation's capital.

"This is the national capital and it deserves to have the very finest performing arts groups the country can put forth," said Arena's Fichandler. "There are a lot of national corporations that should be doing much more here. They have their lobbyists here, they benefit from bringing clients here, but they seem to take a very narrow view of what their responsibilities are."