Fairfax County, which suffers from a severe shortage of low-cost housing, risks losing more than $2 million in federal funding to build 30 units for low-income families because of problems finding a site for the project, county officials said yesterday.

Fairfax, one of the most affluent counties in the nation, has received three extensions from the U.S. Department of Housing and Urban Development for approving the politically unpopular subsidized housing. The county now faces a Sept. 30 deadline to find a location for the 30 three-bedroom garden apartments or town houses.

County officials say that they have considered about 250 vacant properties but that none has been suitable for the units.

"People want to help their fellow man . . . but they don't want to do it in the neighborhood," said county Supervisor Thomas M. Davis III, Republican chairman of the county board subcommittee on housing. "I don't think anybody's enthusiastic to pick this thing up."

Local officials and community leaders have warned that the dwindling stock of low-cost housing in the county could harm the local economy as blue-collar laborers and entry-level professionals seek work elsewhere because of the county's high cost of living.

Despite those concerns, many members of the Fairfax County Board, who have the final authority over such projects, have shunned subsidized housing proposed for their districts.

Supervisor Elaine N. McConnell, a Republican who represents the Springfield District in the fast-growing western part of the county, said there are already too many apartment buildings planned for her district. "I don't want to be the one who goes back to the voters and says, 'Hey, could we put in a few more?' "

Federal funding of public housing has grown increasingly scarce under the Reagan administration, and the HUD funds now offered to Fairfax could be among the last such grants available.

The House approved legislation Thursday that would halt nearly all construction of new public housing.

"There is a risk that if we can't find a site that we would lose" the money, said Walter D. Webdale, the county's director of housing and community development.

More than 4,000 families in Fairfax are on a waiting list for subsidized public housing, and the county has stopped accepting names for the list.

To be eligible for the subsidized housing that HUD has offered the county, a family of four would have to have a maximum annual income of about $20,000.

Throughout the county, the amount of government-subsidized low-cost housing is declining dramatically. In addition, privately owned low-rent apartment buildings are being sold and upgraded.

According to Sharon Kelso, executive director of United Community Ministries Inc., an activist group involved with housing issues, the county is on the verge of losing about 400 units of low-cost housing because of the imminent sale of apartment buildings now subsidized by the federal government.

County officials are considering a move that would further reduce low-cost housing by selling two-thirds of the Woodley-Nightingale Trailer Home Park off Richmond Highway (Rte. 1) to raise money to renovate the remaining one-third of the park.

Some members of the county's Redevelopment and Housing Authority board, which oversees the mobile home park, said at a meeting Thursday night that they favor selling the entire park, which houses 260 trailers, and using the income to relocate the displaced residents.

The county "seems to be doing everything within their power to get rid of the low-income housing that's currently available," said Kelso. "It's being done under the guise of redevelopment but it's a misguided sense of what's needed to redevelop certain areas."

Mary Sue Lyons, chairman of the Fairfax County Housing Coalition, a low-cost housing advocacy group, said, "It's hard for me to believe that there isn't a place anywhere in a county that's 400 square miles to put these 30 units."

County officials say they are sincere about trying to find a site for the 30 new units that HUD would finance. Officials said the shortage and high cost of land in the county, as well as zoning obstacles to the project, have hampered their efforts.