But it would remain in force longer if all losses to the bailout fund, the Troubled Asset Relief Program, were not recovered after a decade. The Treasury now projects that the losses from the $700 billion loan program, which was created in October 2008, could reach $117 billion, about a third of the loss that it projected last summer  an improved forecast that reflected the renewed strength on Wall Street.

The White House said that collecting $117 billion would take about 12 years, but Treasury officials said losses were likely to be smaller. Still, with postrecession deficits at levels unseen since World War II and Wall Street never broadly popular, the pressure on a future president and Congress to keep the tax in place is likely to be substantial. Administration officials did not outline any provision for having the tax expire once all the money is recouped.

Big banks object that most of them already have repaid the government with interest. The administration, anticipating that argument, called its tax a “financial crisis responsibility fee” aimed at those institutions whose risk-taking caused the problem in the first place.

The losses from the bailout fund are expected from money paid to rescue Chrysler and General Motors and the insurance giant American International Group, and from a program to help homeowners avert foreclosures. But big banks shared in the A.I.G. bailout, splitting about $60 billion to receive full repayment for their financial trades with the company.

Mr. Obama’s economic team began seeking a bank fee last August, even as the administration  and in particular the Treasury secretary, Timothy F. Geithner  was being attacked by critics on the left and right as too cozy with Wall Street. Criticism picked up last year after Mr. Geithner opposed an effort by the European Union to impose a global tax on financial transactions.

Mr. Geithner said such a tax would be passed through to customers. The administration now argues that big banks will not be able to pass on the costs of its levy without risking a loss of market share to rivals that are not subject to the tax.