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Mr. Obama said the debate is likely to depend on whether Republicans get pushback from constituents at home for their stand.

House Republicans at times have tried to argue that they aren’t breaking new ground.

They circulated a list of 15 emergency spending bills over the past decade or so that have had offsets included — though many of those were war-spending measures or other types of emergencies, not natural disasters.

Those that did include disaster money did not make the explicit trade-off between the extra funding and cuts elsewhere.

Earlier this month, Sen. Tom Coburn, Oklahoma Republican, proposed offsetting the disaster money by cutting duplicative programs identified by Congress’ investigate arm, the Government Accountability Office, in a report issued this year. A few months ago, a similar amendment, co-sponsored with Sen. Mark Warner, Virginia Democrat, garnered 64 votes.

This time, it got 54 votes — six shy of the 60 needed for passage — and nearly a dozen Democrats who voted for it last time voted against it this time, including Mr. Warner.

Mr. Warner’s spokesman said the dollar amount was higher this time and that he didn’t want to entangle disaster money in a political fight over offsets.

Democrats could have offered their own offsets, such as some of the targeted tax increases President Obama proposed this month in his jobs-stimulus package. But they chose not to.

The fight has become personal at times. Sen. Mary L. Landrieu, Louisiana Democrat and chairwoman of the spending subcommittee that oversees FEMA funding, called out Republican lawmakers who have asked for clean-energy funding in the past, but who now back cutting funding to pay for the disaster money.

Ms. Landrieu warned that she will hold them to their own standard.

“I’m just saying I am not going to forget this vote because I chair this committee,” Ms. Landrieu said. “And if you vote to require an offset and another storm hits your state, then it is going to be the responsibility on your shoulders to tell your people, ‘I’m sorry I can’t help you until I go to Washington and find an offset.’ “