Image: Phoenix © Mark Downey, Lucid Images, Radius Images, Getty Images

For sale or rent by distressed owner: 248,000 homes. That's how many residential properties the U.S. government now has in its possession, the result of record numbers of people defaulting on government-backed mortgages. Washington is sitting on nearly a third of the nation's 800,000 repossessed houses, making the U.S. taxpayer the largest owner of foreclosed properties.

With even more homes moving toward default, Fannie Mae, Freddie Mac and the Federal Housing Administration are looking for a way to unload them without swamping the already depressed real-estate market.

Trouble is, they haven't figured out how to do that. The government admitted as much in August, when Fannie, Freddie and the FHA issued a joint plea to the public for ideas about how to solve the problem. (Give it your best shot: You have through today, Sept. 15, to submit ideas to reo.rfi@fhfa.gov in a .pdf format. Click here for more details.)

"They're stuck," says Karen Shaw Petrou, a managing partner of Federal Financial Analytics, a Washington, D.C., consulting company that advises banks and other clients on government policy. "They don't know what to do."

Since the 2008 financial collapse, the government has spent billions of dollars trying to extricate borrowers from high-cost loans, aid delinquent homeowners and stabilize neighborhoods. The results have been disappointing.

The Obama administration's signature loan-modification program has helped about 657,000 homeowners -- far short of its goal of 3 million to 4 million. The program was a victim of its complexity and its inability to cope with overwhelming demand. Many families hit hardest by the housing downturn are concentrated in states that are having the most difficulty recovering from the recession, including Florida, Ohio and Nevada.

The government's call for ideas is a sign it is deluged with repossessions, commonly known as real-estate-owned properties, or REOs. "It's almost like having the captain of the Titanic go on the public address system and say, 'Does anybody have an idea?'" says Mark Wiseman, a former director of Cleveland's foreclosure-prevention program. "It's not a confidence builder."

Fannie Mae, Freddie Mac and the FHA made progress in the first half of this year, reducing their combined backlog from 295,000 single-family homes in December to about 248,000 in June, according to the Housing and Urban Development Department. The nation's total number of repossessions also fell during that period, from nearly 981,000 to about 817,500. The government's share has remained steady at around 30%.

In coming months, however, as lenders and the courts clear up the "robo-signing" scandal that slowed new disclosures, the number of government-owned properties will likely grow. More than a fifth of the 3.65 million homes for sale at the end of July were foreclosures, according to RealtyTrac, a housing-data provider.

"It isn't necessarily our preference that FHA is going to itself continue to hold these properties," says FHA acting Commissioner Carol Galante. "We want to move homes through the system so we can recover."

The agency has to be careful as it goes about it, Galante says. "If you're putting too much through that system, you are helping to drive down prices." That's especially true in regions congested with government properties.

Shielding the market from a flood of government homes might be good for property values and the economy. It's not such a great deal for taxpayers, who bear the costs when government-guaranteed loans go bad and who pay for maintenance on vacant homes the feds take over. One idea the administration is exploring: allowing Fannie, Freddie and the FHA to keep an ownership stake in the properties by converting them to rentals in partnership with private investors. When the market recovered, the government would sell the homes for more than it could get now and not risk glutting the market. Structured properly, such joint ventures could reduce the impact of foreclosures on struggling neighborhoods.

It's not at all clear whether that would work on a large scale. The government would have to spend money to bring the rental properties -- many of them old and dilapidated -- to code, pay still more to insure the rentals and build a bureaucracy to manage and maintain them. Even if it does all that, there might not be people willing to move in. In parts of Cleveland and Detroit, for example, some houses are stripped and vandalized the minute they're vacant.

Click here to become a fan of MSN Money on Facebook

"Some of the neighborhoods, you can't move into,'' Wiseman says. "There are so many empty houses, it's just not safe."

In places like that, it's sometimes difficult to persuade people to stay in their houses. Freddie Mac allows occupants of foreclosed homes to remain on month-to-month leases until the homes are sold. Few do, spokesman Brad German says. "People prefer to take cash for keys and move on."

This article was reported by Lorraine Woellert and Clea Benson for Bloomberg Businessweek.