Representatives of the Fairfax County Board of Supervisors and the county's embattled housing authority approved a sweeping plan last night that would strip the agency of much of its power and turn it over to the supervisors.

The action was spurred by years of bickering and political rivalry between the agency and the supervisors, some of whom have sometimes been unsympathetic to public housing for low-income families in the wealthy suburban county.

The proposal, which must be approved by the full Board of Supervisors and the full Redevelopment and Housing Authority before it can be implemented, would give the board tighter control over the agency's spending and programs. It also would turn over day-to-day authority over the agency to County Executive J. Hamilton Lambert rather than leaving it in the hands of the housing authority's director.

"The authority is being put back in the hands of the elected officials in Fairfax," said Annandale Supervisor Audrey Moore, a Democrat, adding that the agency will become an advisory board to the supervisors.

"There isn't a blasted thing in the world we can do," said Carl A.S. Coan Jr., chairman of the housing authority and an outspoken opponent of the proposal. "We have all the responsibility, yet we have no authority."

Some county officials said the proposal is only one part of a major overhaul in the way the county provides services for low-income residents, a move away from government-financed assistance toward more private-sector funding. It is a policy that some officials predict now will be pushed harder by the new Republican majority on the board.

"The authority has not been keeping up with the times," said Republican Supervisor Thomas M. Davis III, who represents the inner-county Mason District. "Public financing is no longer the answer in the Reagan era."

Even before Republicans took control of the board in this fall's elections, Democrat Moore said supervisors, who appoint members of the housing authority, had been slowly stacking the authority in favor of a move to turn over more control to the board.

Moore's appointee to the authority, David G. Russell, said he believes the agency will accept the plan: "The authority has always bothered me. We're not responsible to anybody but the supervisors."

If the authority vetoes the plan, Davis warned that supervisors will replace recalcitrant members of the authority.

Some supervisors and officials of the housing authority have been feuding for almost a decade. Supervisors have accused the agency of sloppy handling of the millions of dollars that are funneled through it each year.

Earlier this year, the county board paid a private consultant almost $100,000 to evaluate the agency and its programs. The report found validity in many of the allegations of poor management and lack of clear-cut goals for serving the county's low-income residents.

The authority operates myriad programs for low-income residents, including public housing, rent subsidies, housing construction and construction of new roads and sewers. It also administers numerous loan and grant programs.

Some housing authority officials blame their long-running battle with some supervisors on the unpopularity of programs for the low-income residents among supervisors who represent one of the most affluent counties in the nation.

They contend that the agency is one of the most complex in the county and that lines frequently are blurred between the federal, state and county programs the authority is expected to coordinate.

The authority has been criticized recently for what some supervisors have called excessive use of private lawyers. The authority uses more outside lawyers -- 12 firms this year -- than any other agency the county. It's budget for private lawyers fees this year was $300,000.

But housing authority officials say the agency's heavy dependence on outside lawyers is the county's fault. They say supervisors repeatedly have turned down their requests for money to hire more in-house counsel.

Under the proposal outlined last night, the county attorney's office would assume control over the agency's legal matters.