The tax on banks, insurance companies and brokerages with more than $50 billion in assets would start after June 30 and seek to collect $90 billion over 10 years, according to a senior administration official who briefed reporters late Wednesday.

But the levy but would remain in force longer if all losses to the bailout fund, the Troubled Asset Relief Programme, are not recovered after a decade.

Administration officials now say that the losses from the $700 billion loan program created in October 2008 are likely to be about $117 billion, which is about a third of the losses that the government projected last summer — an improved forecast that reflects the strength of the recovery on Wall Street, even as Main Street struggles.

“The goal of this proposal is to ensure that those major financial institutions who were the most significant contributors to the financial crisis, and who have been the most significant beneficiaries of the extraordinary public policy efforts, take responsibility to ensure that all of the costs of TARP are not added to the deficit or passed on to next generations through an increase in our national debt,” said the official, who spoke on condition of anonymity in advance of the president’s announcement.