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In the House, Rep. Barney Frank, Massachusetts Democrat, is taking a more straightforward route to save the homeowner-aid programs that he helped create. He is resurrecting the idea of levying a tax on the biggest banks to raise money for the loan modifications.

But the White House and Democrats have a fight on their hands no matter which course they take.

Four Republican state attorneys general from Virginia, Texas, Florida and South Carolina objected last week to attempts to extract money from banks for a broad loan-modification effort as part of a deal intended to settle charges that the banks bungled the paperwork in thousands of foreclosures and may have wrongly evicted a few people from their houses.

The Republican attorneys general said any penalties extracted should be limited and aimed at remedying “unlawful conduct at issue in the investigation,” such as improperly signed foreclosure documents, in a letter to Iowa Attorney General Tom Miller, who is leading the 50-state investigation.

Republicans on Capitol Hill are raising the same issue. Sen. Richard C. Shelby, Alabama Republican and ranking member of the Senate Banking, Housing and Urban Affairs Committee, called the proposed settlement a “regulatory shakedown” of the banks.

The settlement offer was put together last month by state attorneys general in conjunction with the new Consumer Financial Protection Bureau established by last year’s banking reform law, the Treasury Department, Federal Reserve and Federal Deposit Insurance Corp.

Mr. Shelby questioned the “strong-arm tactics” being used to try to coerce the banks into the settlement. The four targeted banks - which process payments on most of the country’s mortgages and send them to investors who have purchased the loans - also are signaling they will fight the proposal.

Bank of America chief executive Brian Moynihan questioned how banks could be expected to help some homeowners but not others, and said the proposal would only encourage more people to walk away from their mortgages.

“When you start helping certain people, and don’t help other people, it’s going to be very hard to explain the difference,” he told investors earlier this month.

Mr. Hendler said that requiring banks that are servicing the loans - but didn’t originally make them all - to write down principal and absorb losses “raises sensitive issues of moral hazard” and may prevent banks from taking on the role of servicing mortgages that were made by other lenders in the future.

Many of the bad loans were made by defunct lenders such as Countrywide and Washington Mutual, which were taken over by JPMorgan, Bank of America and the other big banks in 2008 as part of mergers arranged by the government to prevent an implosion of the banking system during the financial crisis.