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Mr. Cornyn said Republicans’ willingness to impose a partial shutdown should be a warning to President Obama, who has said he will not negotiate changes to spending with Republicans in exchange for another increase in the government’s borrowing ability.

But even if he won’t negotiate on the debt ceiling, Mr. Obama will be unable to avoid negotiations over the annual spending bills, which expire March 31.

In the wake of the tax fight, some analysts have said Mr. Obama now has a weaker hand because the tax-rate issue is off the table and the fight now is on Republican issues of spending and debt.

But House Minority Leader Nancy Pelosi, California Democrat, said Friday that she thinks Mr. Obama holds a strong hand after the debt fight.

She also said Mr. Obama should consider bypassing the debt fight altogether and claim the constitutional ability to make good on the debt regardless of the ceiling set by Congress.

“I would do it in a second,” she told reporters.

House Speaker John A. Boehner of Ohio, talking with fellow Republicans behind closed doors Friday morning, didn’t go as far as his Senate colleagues in warning of a shutdown, but he did say Mr. Obama will have to talk spending cuts in exchange for a debt limit deal, according to a source in the room.

“The debate is already under way,” he said.

• Valerie Richardson contributed to this report.