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Boehner mentioned no numbers, saying it would depend on proposals by FEMA and the White House, and damage totals from states, and said he didn't know if it would be taken up this year or next, King said in an interview.
Boehner's spokesman did not respond to requests for comment on his conversation with King or his position on a supplemental appropriation.
Boehner's conversation with King is the first indication that the speaker accepts the need for new money, and his backing could remove a hurdle in the GOP-controlled House for expanded federal assistance to New York and other storm-damaged states, congressional aides said.
Last year, the House held up funding for a depleted FEMA as it sought to send aid to New York and other states flooded by Tropical Storm Irene, demanding that any new money first be offset by cuts to other programs.
King said Boehner said the new supplemental funding for Sandy would not require spending cuts elsewhere, but it is not clear fiscal hawks would go along with that promise. Sen. Tom Coburn (R-Okla.) would seek to require offsetting cuts before adding new FEMA funds, a Coburn aide said Tuesday.
"It is critical that this budget amendment be submitted as soon as possible so critical resources can reach impacted communities by the end of the calendar year," the senators said in the letter.
The senators also asked President Barack Obama to "consider increasing the federal share for public assistance for affected states."
The White House did not respond to requests for comment on the letter.
But before the money can be appropriated, the affected areas must produce solid estimates, and an itemized list of needs.
So far, Gov. Andrew M. Cuomo has released a $30 billion estimate for New York, but it includes few specifics.
With Robert Brodsky
and Paul LaRocco