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The parliamentary situation is complex, but one option for Senate Democrats is to wait until late Monday and move to table both House proposals. If the Democrats, as expected, can muster at least 51 votes to kill the proposals, the original bill will go back to the House just hours before the midnight deadline.

House Republicans then would have to scramble to pass yet another proposal.

Majority Whip Kevin McCarthy of California, the third-ranking House Republican, told “Fox News Sunday” that other ideas were in the works and vowed that the government would not shut down — but he gave no specifics.

Democrats eagerly fundraised off the showdown, which coincides with a major quarterly fundraising deadline. The Democratic Congressional Campaign Committee begged supporters to pony up for their Obamacare Rapid Response Fund. Mr. Obama’s own campaign arm, Organizing for America, pleaded for money it said it would use to shift the focus from Obamacare to immigration.

Mr. Reid has said he will not accept any conditions on the spending bills, while House Speaker John A. Boehner, Ohio Republican, insists on having strings attached.

The Republican strategy relies on unity, however, and that is beginning to fray.

Sen. Susan M. Collins, Maine Republican, released a statement rejecting GOP insistence that Obamacare be tied to the spending bill.

“I voted against Obamacare and have repeatedly voted to repeal, reform, and replace it, but I disagree with the strategy of linking Obamacare with the continuing functioning of government — a strategy that cannot possibly work,” she said.

She said she wanted all sides to pass a short-term spending bill to carry the government beyond the midnight Monday deadline, and then asked Mr. Obama to meet with her and a small group of Senate Republicans who had been trying to work out a long-term debt deal.

With a shutdown looming, avoiding blame has become a key goal in the strategies of both Democrats and Republicans, with memories of the 1995 and 1996 shutdowns still vivid in many lawmakers’ minds.

In those instances, congressional Republicans shut down the government in their battle over spending with President Clinton. They suffered for it at the polls, losing seats in two successive congressional elections.