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“It’s safe to say there’s no clear consensus yet on how best to design a new system,” Mr. Geithner said. “But this administration will side with those who want fundamental change.”

Proposals from liberal groups and Democrats have centered on the government providing a kind of last-resort insurance for the mortgage market just as it provides last-resort financing for the terrorism-insurance market.

Banks that originate and pool the mortgages would have to sustain losses and exhaust their resources before the government would step in and backstop the securities. And investors and borrowers would have to explicitly pay upfront for the government’s guarantee.

Democrats point to Standard & Poor’s opinion and statements by major investors such as Pimco’s Bill Gross, co-founder of the world’s largest bond fund. He said he is reluctant to buy mortgages that don’t have a government guarantee unless borrowers have made substantial down payments of 30 percent or more or are willing to pay significantly higher interest rates.

The criteria Mr. Gross laid down shows how difficult it would be for borrowers to obtain loans in a privatized market, since many today continue to have trouble coming up with even the standard 10 percent to 20 percent down payments and have flocked to federal housing programs that allow smaller upfront investments.

Moreover, the banking industry has been warning legislators that wholesale elimination of Fannie and Freddie could result in the disappearance of the 30-year mortgage — a feature that was not available for home purchases before the creation of the two enterprises in the 1930s.

The loss of 30-year loans would hurt borrowers and might change the structure of the housing market, in which prices have risen solidly over the years based on the availability of such low-cost, long-term financing.

In the future, borrowers would have to choose from an array of riskier and more expensive options such as 10-year loans and adjustable-rate mortgages that do not guarantee the low, steady monthly payments provided by 30-year loans.

Heeding warnings from bankers, real estate agents and developers who contributed heavily to last fall’s Republican campaigns, some GOP leaders are signaling a willingness to work with Democrats on a bipartisan plan that would continue to provide a limited federal guarantee fully covered by fees to the government.

But others are expected to trumpet their criticisms of Fannie and Freddie in hearings next month before several House committees that are sure to be crowd-pleasers for conservative constituencies.

Rep. Darrel Issa, chairman of the House Oversight and Government Reform Committee, gave a taste of what is to come Monday when he pointedly questioned why the government has paid more than $160 million in legal fees to defend former Fannie and Freddie executives.

“At a time of runaway federal deficits and 10 percent unemployment, it is extremely distasteful for the American taxpayers to be forced to pay the legal bills,” the California Republican said, charging that the former executives were “central players in the financial crisis.”

In an attempt to make good on Republican promises to do away with the enterprises, one conservative plan drafted by analysts at the American Enterprise Institute would eventually eliminate them and try to restart the private mortgage market by gradually lowering the limits on the size of loans that Fannie and Freddie can guarantee. The limits currently range up to $729,000 in major metropolitan areas.

“There is near-universal agreement that Fannie and Freddie should not continue” as they have for the past three decades, said Alex J. Pollock of AEI, though there are “numerous conflicting ideas” on how to replace them.