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In that fight, the issues were spending and deficits. This year’s impasse is tied to the health care law.

The law’s state-based health exchanges are scheduled to open Oct. 1, coinciding with the next fiscal year, and many conservatives say the spending bills are the last, best chance to halt the law before it takes full effect.

Republican leaders have been skeptical, and last week they tried to offer their members a symbolic vote that all sides knew would fail. That sparked a rebellion within the party, so they came up with a two-step strategy.

Democrats urged Republicans to drop their focus on the health law.

“They’re obsessed with the constitutional law that’s been in effect now for four years — declared constitutional by the United States Supreme Court,” said Senate Majority Leader Harry Reid, Nevada Democrat.

The spending fight is about more than health care, however.

The House bill would spend less than Democrats want and would leave in place the “sequester” budget cuts that stemmed from the 2011 debt deal.

Republicans are trying to decide where they have the most leverage.

“I like the idea of a fight on the debt ceiling better” than on the budget, said Rep. Todd Rokita, Indiana Republican.

Mr. Rokita said the American public historically associates government shutdowns with the GOP, particularly after the party’s fights with President Clinton in the mid-1990s.

This time, Mr. Obama would be more likely to bear the responsibility for upsets in the economy that are tied to the country’s debt limit, Mr. Rokita said.