Washington's new Housing Finance Agency, created to help increase the supply of homes here for poor and middle-income families, has become embroiled in a squabble with the city housing department that could cripple HFA almost before it begins to work.

On the surface the fight is over which agency gets the authority to raise housing construction money by selling bonds. But the dispute goes to the question of who benefits when city agencies compete with each other to make financing deals, and its outcome, HFA officials say, could limit at least for a time the number of new homes their agency can help build in Washington.

The battle began last March, when HFA officials asked city administrator Elijah Rogers to give them exclusive rights to sell tax-exempt bonds, taking away the city housing department's authority to use that tool for raising construction money. The request was prompted by HFA's desire to be the source of any city financing aid for a series of proposed projects that would provide a total of about 700 new low- and moderately priced apartments.

The projects are important to HFA because they would count against its legal obligation to insure that at least half the units it helps build are affordable for low and moderate-income residents.

They were particularly attractive because the federal government has already pledged to pay portions of the rent for the low-income tenants in the new buildings. Nearly every new apartment project built here in the last two years has depended to some degree on that federal program, known as "Section 8" after the section of the law that created it, which will end Oct. 1 as part of the Reagan administration's budget cutbacks.

When Section 8 goes, analysts from every side of the housing industry agree, building of apartments and homes priced for low-income persons is likely to come to a virtual halt.

Eventually HFA and agencies like it are expected to find ways to fill the gap, but HFA chairman Floyd Hyde fears unless his agency can quickly begin some low-income projects, the legal requirement for matching low-income and other units will serve to stop HFA from building any units until some replacement for the federal program can be found.

"As the federal assistance changes," he said, "how do we accumulate enough low-priced units to get our 50-50 mix that is required by law? If the city uses the housing department , it would undercut our ability to function."

HFA did not want developers to have a choice of using either their financing or the housing department's for the projects, because they would be likely to lose in such a competition. HFA proposes to use procedures that will give it control over use of proceeds from bond sales, but those procedures will also mean more expenses for developers.

"It makes no sense to have two agencies that developers can play off against each other," Hyde said. "It's a bad policy."

However, not everybody sees the situation that way. HFA's request to Rogers sparked an outcry from the housing department, which felt its turf was being threatened, and from area developers unhappy over the prospect of being forced to deal with a single new agency that would offer them a less attractive deal.

Housing director Robert L. Moore said HFA was trying to take away projects "we had come a long way with." Moore, who is a member of the HFA board, disagrees with Hyde's contention that the agency needs the units in order to remain active. "I say they don't need the units," he said. "There are other ways to meet the requirements."

"The HFA looked like an eleventh-hour interloper that was changing the rules," said one source closely tied to the developers in the dispute. "And the developers did not want HFA to have a monopoly" because they wanted the ability to shop between the agencies for the best financing deal.

"The developers screamed murder," said Moore. "They raised Cain."

"We can understand why HFA is doing this -- they're fighting for their existence," said one developer. He added, however, that an end to the kind of funding offered by the housing department would be too expensive for them to tolerate readily.

For the developers the question of which agency issues bonds can mean a difference of thousands of dollars in costs. Under HFA's plan, the agency would keep control of the bond proceeds and any interest they earn before the time that the bonds must be redeemed, and could use that interest to defray costs of the bond issue or to pay other agency costs.

Under the federal regulations followed by the housing department, the bonds are actually issued by non-profit corporations formed in connection with each project, and developers can control the interest earned by bond proceeds.

Under federal supervision, they can use that money to pay costs of the bond issue or to retire the bonds, a procedure that cuts their costs and could increase their equity in the projects.

Moore contends developers should be allowed the same choice in the District that they are in other jurisdictions, where state-run housing finance agencies exist alongside local housing departments. He accuses the HFA of "playing brinksmanship with needed housing," while HFA argues that the housing department has been saving the developers money while hampering HFA's future operations.

HFA needs the income from fees charged to developers plus investment income from bond proceeds if it is to pay its own operating costs, as required by law. Operating costs for the housing department, like other city agencies, are paid from the city treasury. HFA officials contend that taxpayers would benefit from having their self-supporting agency handling the bulk of housing financing deals.

Both Moore and Rogers said the city supports HFA and would pay for it out of public dollars if the agency was unable to generate enough money from its development deals for its operations.

The fight between the two agencies is not an unusual one. Sources familiar with similar programs elsewhere said competition often erupts between state housing finance agencies and local housing departments because developers prefer the financing offering by housing departments. But they say that state finance agencies often can operate freely in rural areas while housing departments concentrate on cities. In Washington, the competition is tougher because both agencies operate in exactly the same geographic area.

After hearing arguments from both sides, Rogers denied HFA's request and told the two agencies to work out their differences. HFA gave up the fight over the immediate projects, and the housing department is in the process of arranging the sale of those bonds.

HFA sold its first public bond issue last week in connection with another project, raising $57.4 million to finance the construction or renovation of 257 apartments for low- and moderate-income families. The housing department has authorized the sale of more than $100 million in tax-exempt bonds since Moore became housing director in 1979. Those bonds financed the construction of more than 1,200 apartments for families earning generally less than $20,000 a year.

Council member and mayoral candidate Charlene Drew Jarvis (D-Ward 4), who chairs the council's housing and economic development committee, called the competition between the two agencies "unfortunate," and said financing the 700 units could have helped HFA get off the ground. She said if steps are not taken to protect HFA's ability to function adequately in the future, the council will step in.

"HFA is one of our best opportunities to develop affordable housing," she said, "And I will not see it decline because of a turf battle."