After a four-month search that has included a trip to California and a string of five-hour dinners at their own headquarters, the Fairfax County School Board is nearing a decision on selection of a new superindendent to run the county school system.

Under state law, the selection for the $55,000-a-year job must be made by Nov. 28, 120 days after the departure of former superintendent S. John Davis. He quit the Northern Virginia job to take a lower paying position as state school superintendent in Richmond.

Fairfax school board members said yesterday they have narrowed the field from 105 applicants to three finalists. But no one on the board is saying who the final three are.

"We promised we wouldn't reveal their names, and we're going to stick to that," said board chairman Rodney F. Page. "If we told who was applying, we would discourage qualified people. I don't think there's anything to gain by parading the candidates in front of the public and letting the special interest groups debate their merits."

Other board members said the three finalists come from California, Alabama, and Florida. All are men in their 40s or early 50s who carry doctorate degrees and already head smaller school systems, one in a city and two in suburbs.

"Some people tease us that we want a saint," said board member Ann P. Kahn, "It's not that, but we are looking for someone who will continue the excellence we have, and not turn things upside down."

With 128,000 students, Fairfax now is the largest school system in the Washington area and the 10th largest in the United States. It is in the middle of a dispute with a teachers group over salaries, with many teachers "working to the rule," rather than striking. Virginia localities are prohibited from having collective bargaining agreements with public employe unions.

Despite its size, Fairbax has avoided the divisive battles that mark school board politics in Washington, Montgomery, and Prince George's counties. Its latest superintendent, Davis, served for nine years, an exceptionally long time for a school superintendent to survive during the 1970s.

Several board members said one reason for the relative calm on the Fairfax school board is that it is appointed by the county supervisors, not popularly elected.

"Things are less political," said board member Gary L. Jones. In going about the business of selecting a new superintendent the board has tried hard to keep them that way.

With a search budget of $25,000, the board placed job advertisements in dozens of magazines and newspapers across the country, and hired two consultants to evaluate the applicants.

After meeting with community groups, the board established its own formal criteria for the job in addition to the strict education requirements for all superintendents established by the Virginia State Board of Education.

The two consultants, Richard Wynne of the University of Pittsburgh, and Carroll Johnson, a retired professor at Columbia Teachers College, matched the candidates against all the requirements.

They came up with seven semifinalists for the board to consider: all men; six white, one black; one from Fairfax, two from elsewhere in the metropolitan area, four from elsewhere in the country.

The board invited each of them to dinner at its headquarters, spending about five hours with each candidate.

Afterwards there was a long meeting during which the board reached a consensus cutting the list down to the three finalists. Four or five board members then visited the communities where the men work, interviewing officials, teachers and community groups. What's more they succeeded, board members said, in keeping the whole project out of the newspapers.

Now the finalist are visiting Fairfax again to ask their own questions. Within the next 10 days, board members said, they hope to get together and make a final decision. They said they do not plan to consult with county supervisors or anyone elese in Fairfax about the specific men involved. c

"We have the legal responsibility to choose a superintendent," said Kahn, "and we're going to do to. . . . If it works out well, then we'll get the credit. And if it doesn't, then we'll be the ones who are responsible."