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“What we’re seeing in our polling and in the public polling is that Republicans are in fantastic shape when they’re talking about Obamacare, and that includes with independents, right up until the point where you start talking about the government,” he said. “At that point, it becomes more of an ideological issue where you start to lose some independents, and some Republicans, too.”

Mr. Haskins said if the GOP does precipitate a shutdown over the debt limit, “it’s going to really hurt the party.”

The Republican-led House and the White House are far apart on the levels to set government spending in the new fiscal year, which begins Oct. 1. Many negotiators expect a short-term measure to keep the government operating past Oct. 1.

But Mr. Obama has demanded a debt-ceiling increase with no conditions, a proposal that some Republicans oppose. House Republicans are pushing for cuts and budget reforms as conditions for raising the national borrowing limit.

There are only nine working days for the House this month, but Mr. Haskins said he expects a last-minute, short-term deal to keep the government running.

“They’ll have a continuing resolution and they won’t close the government and they’ll reach some kind of compromise on the debt ceiling,” he predicted. “How they get there, God only knows.”