The Obama administration is considering further actions to strengthen the housing market, but the bar is high: plans must help a broad swath of homeowners, stimulate the economy and cost next to nothing.

One proposal would allow millions of homeowners with government-backed mortgages to refinance them at today’s lower interest rates, about 4 percent, according to two people briefed on the administration’s discussions who asked not to be identified because they were not allowed to talk about the information.

Administration officials said on Wednesday that they were weighing a range of proposals, including changes to its previous refinancing programs to increase the number of homeowners taking part. They are also working on a home rental program that would try to shore up housing prices by preventing hundreds of thousands of foreclosed homes from flooding the market. That program is further along — the administration requested ideas for execution from the private sector earlier this month.

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