The long running controversy in the Fairfax City over whether Fairfax County should continue to run the city's schools, or whether the city should do that job itself, haunts local politics "like a serpent under the water," in the words of one city councilman.

The serpent stirred mightily for several months earlier this year, until the city school board last month rejected a proposal to hold a referendum on whether the city should run its own independent school system.

A series of other events suggest that the prospect for an independent school system in the next few years is poor. Even so, the stirrings continue.

Already, the school board has implicity acknowledged it is unlikely that it will establish an independent school system, by replacing a full-time $33,000-a-year superintendent with one who will be paid $12,000 for 24 hours of work a week.

Later this month, the City Council will consider the reappointment of the school board chairman, George Hamill, a staunch advocate of an independent school system. The decision is likely to reflect the Council's attitute on the independent school issue and a recent boardcouncil conflict. The Council will also fill the position left open by the resignation of John Russell, another independent school system supporter.

Meanwhile, the city is preparing to renegotiate all city-county contracts - including those affecting fire service, water and sewer service and courts as well as school - and several Council members have told the school board to stay out of the early negotiations.

Some school board members have protested that the Council cannot properly exclude them from the negotiations; the school board attorney will report the board on this question next week. But it is clear that eventually both city and school board will have to grapple with the tough problem of exactly what they would like to change in the school contract.

The two Fairfax have an unusual school setup. The city has its own school board and school superintendent, as required by its charter and state law. But since Fairfax City became a city in 1961, the city school board has simply contracted for the county to run its schools. So while the city owns the school buildings and retains nominal authority through the board and superintendent, the real control of its schools rests with the county. There are now seven schools in Fairfax City with 5,000 students.

When Fairfax City changed from a town to a city, it was apparently expected that the city would run its own schools, after a short period of countyrun schools under a contract. The board hired a full-time superintendent and geared up for a transition that never happened.

Last month the school board bowed to reality and accepted the resignation of George G. Tankard, the school superintendent since 1971 who had worked at his job full-time and who was making a salary in excess of $33,000.

The new superintendent, Wayne H. White, took over on July 1. He will spend only 24 hours a week at the job, for $12,000 a year. But significantly, his four-year contract has a yearly cancellation clause to enable the school board to hire a full-time superintendent, in case it wants to start up an independent school system within 18 months.

Clearly, the issue of an independent school system is on the mind of public officials in Fairfax City - even though the same school board that put a termination clause in White's contract refused last month to recommend a referendum on the subject.

"It's not dead. You don't knock a good idea down. While they've postponed it, they haven't killed it," said Mayor Nathaniel E. Young.

"It never goes away. It's kind of life a serpent under the water. Every election time it comes up. How it surfaces, I don't know. What causes it to surface, I don't know," sighed City Councilman Frederick J. McCoy.

A citizen task force's study of the city school system was the impetus for much of the activity earlier this year. Formed in mid-1976, the group surveyed citizens about the schools and reported on school facilities, programs and the advisability of having an independent school system.

When the task force issued its reports this spring, the school board focused on the section covering an independent school system, holding two public hearings and eventually voting not to recommend to the City Council that a citywide referendum be held on whether the city should run its own schools.

On June 6, when the school board voted, three of five school board members favored referendum. But because of disagreement over minor aspects of two motions favoring an independent school system and a referendum, the matter lost on a tie vote with Hamill abstaining.

Meanwhile, the board had been pursuing a related matter that involved a conflict both with the county and the City Council. Some city officials thought the county had been overcharging the city for tuition, because of a change in th way the county deducted from its overall costs the revenues it received from special sources.

Early this year, the school board decided by a 5-0 vote to send a letter to the county protesting what it considered a tuition overcharge of $1.1 million occuring over several years. But the letter only prompted a return charge from the county that the city had underpaid tuition.

Further, several City Council members thought the school board should not have written to the county without first consulting the Council. McCoy, one councilman who objected to the letter, explained, "We pay the bills - they just submit the bills to us. It's more our responsibility than it is the school board's responsibility."

The school board, after being reprimanded by Council members at a joint board-council work session for sending the letter, has since had no part in the city-county talks. Young said there was "no attempt to exclude the school board," but that the board would not be brought into the talks until specific school issues are reached.

The school board attorney's report, scheduled to be delivered at next Monday's board meeting, could change the board's recent position of staying out of the city-county preliminary talks. But the real test will be when the school board - which must eventually approve any contract changes - joins the Council in specific negotiations on the school contract.

Councilman Walter Stephens Jr., a candidate to oppose Young for mayor next year, said the move for an independent school system was dead for the next two years because of the city-county renegotiations.

"You have to isolate exactly what you think is wrong with the system - the curriculum, teachers or facilities - and define what you want to do to correct it. Then you have to approach the county school board and ask them to correct it. If the county refuses, and these things are important, then it should be placed on the ballot," he said.

Critics of the county-run city system are unlikely to be satisfied with anything but an independent system. John Russell, an outspoken advocate of an independent system who resigned from the school board after the June 6 vote against a referendum, charged that the current school system graduated many "functional illiterates."

But Russell acknowledged that the recent task force survey and public hearings showed the public to favor retaining the county-run system, and said that was why he resigned.

What happens next depends on the school board attorney's report, the Council's decision on reappointing Hamill and filling the vacancy left by Russelll, and the city-county negotiations.

The schools in Fairfax City are likely to continue to be what Mayor Young acknowledges is a "political football." But Young notes that when parents complain that he has made a political football out of their children, he responds, "Someone else has already done it - the politics of Fairfax County has done it."