Life without Fannie and Freddie is the rare goal shared by the Obama administration and House Republicans, although it will not happen soon. Congress must agree on a plan, which could take years, and then the market must be weaned slowly from dependence on the companies and the financial backing they provide.

The reasons by now are well understood. Fannie and Freddie, created to increase the availability of mortgage loans, misused the government’s support to enrich shareholders and executives by backing millions of shoddy loans. Taxpayers so far have spent more than $135 billion on the cleanup.

The much more divisive question is whether the government should preserve the benefits that the companies provide to middle-class borrowers, including lower interest rates, lenient terms and the ability to get a mortgage even when banks are not making other kinds of loans.

Douglas J. Elliott, a financial policy fellow at the Brookings Institution, said Congress was being forced for the first time in decades to grapple with the cost of subsidizing middle-class mortgages. The collapse of Fannie and Freddie took with it the pretense that the government could do so at no risk to taxpayers, he said.

“The politicians would like something that provides a deep and wide subsidy for housing that doesn’t show up on the budget as costing anything. That’s what we had” with Fannie and Freddie, Mr. Elliott said. “But going forward there is going to be more honest accounting.”