Fellow Republicans say Mrs. Bachmann’s position would influence House colleagues who see her as a reliable measure of the Tea Party zeitgeist.

“Michele is a good barometer,” said Representative Jack Kingston, Republican of Georgia. “The Tea Party really likes her.”

Despite the effort that Mr. Boehner had been putting into structuring a deal that Republicans could support without breaking any no-tax-increase promises, his announcement Saturday evening underscored the political reality that many Republicans in both houses were opposed to increasing the debt ceiling no matter what concessions they might win.

On the presidential campaign trail, Representative Ron Paul of Texas said Republicans should not accept any deal that includes a tax increase, calling it a “ploy.” While Mrs. Bachmann has suggested that virtually no bipartisan deal would be acceptable, Mr. Pawlenty told voters last week that “if it comes to a point where they feel that they must,” he said a balanced-budget amendment was an essential trade-off.

Mitt Romney, a former governor of Massachusetts, has said he would agree to increasing the debt limit only if a deal was “accompanied by a major effort to restructure and reduce the size of government.” Jon M. Huntsman Jr., a former governor of Utah, has said spending cuts must be equal or greater than the value of any debt ceiling increase but told reporters Saturday in Florida: “I have every confidence that cooler heads are going to prevail.”

Nearly three dozen House Republicans and another dozen in the Senate have joined most of the Republican presidential candidates in signing a pledge that they will not vote to support the debt limit increase unless Congress approves a balanced budget-amendment to the Constitution, which is unlikely. (Mrs. Bachmann declined to sign the pledge, because she wanted to include a repeal of the Obama health care plan.)