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“I believe that something like this will bring calm to the American people and will bring needed stability to our financial markets,” said Rep. James E. Clyburn of South Carolina, a leader in the House Democratic Caucus.

But the White House again ruled that option out.

Meanwhile, outside lobbying intensified.

Dozens of Chamber of Commerce affiliates and other business groups signed a letter Wednesday backing the House bill, while the U.S. Conference of Catholic Bishops on Tuesday sent a letter opposing the House bill and urging tax increases and defense spending cuts instead of other spending cuts.

“It requires shared sacrifice by all, including raising adequate revenues, eliminating unnecessary military and other spending, and addressing the long-term costs of health insurance and retirement programs fairly,” the bishops stated in their letter.

No similar letter went out to senators, even though their bill cuts more deeply than the House version.

A spokeswoman for the bishops didn’t return a message seeking comment.

Inside the Capitol, Mr. Boehner and his fellow Republican leaders sought to rebuild support after the CBO on Tuesday said the bill only cut $850 billion, which was $50 billion short of the $900 billion debt authority increase the GOP called for.

On Wednesday, Republicans found new cuts totaling $65 billion.

Mr. Boehner also seemed to win support after it was revealed that staffers for the conservative caucus in the House were targeting fellow Republicans in their attacks over the debt debate.

That anger boiled over in Wednesday’s closed-door meeting, and Rep. Jim Jordan, the Ohio Republican who runs the conservative Republican Study Committee, had to apologize for his staffers’ behavior.

Several aides said they thought the furor over the opponents’ tactics, coupled with the pledges Republican leaders were making to hold votes on the balanced-budget amendment, may have earned Mr. Boehner enough support to pass his bill.

That’s a major reversal from just a day earlier, when Republicans were fleeing from Mr. Boehner’s bill, arguing it didn’t cut deeply enough and surrendered ground from the “cut, cap and balance” bill the House passed just last week, which also insisted on a balanced-budget amendment to the Constitution be passed in both chambers.

The new version only requires that the Senate hold a vote.

Kara Rowland contributed to this article.