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And now they fear by using some of the new spending cuts to replace the sequesters, and still counting that as new cuts for purposes of a future debt increase, he’ll end up double-counting.

“Replacing the sequester does not count as a concession Speaker Boehner can claim that he extracted from the president as far as we are concerned,” said Barney Keller, spokesman for the Club for Growth, a group that opposes big government spending. “Those are spending cuts that were going to happen anyway.

“Certainly, if all you are doing is replacing the sequester, which is already in law, then all you are really doing is giving the president a clean debt-limit increase,” Mr. Keller said.

Tim Phillips, president of Americans for Prosperity, said it looks like they “are clearly backing away from the near-term cuts in federal spending that they promised to the American people back when they did the last debt deal.”

Mr. Boehner’s plan would raise $1 trillion through a combination of higher taxes on millionaires and eliminating loopholes in the tax code. It also calls for $1 trillion in spending cuts.

Mr. Obama, meanwhile, had maintained that any fiscal cliff deal must start with higher taxes on individuals making $200,000 or more a year and families making at least $250,000 a year.

White House spokesman Jay Carney told reporters Monday that Mr. Boehner’s latest proposal did not pass the litmus test for the “balance” between tax increases and spending cuts.

“Thus far, the president’s proposal is the only proposal that we have seen that achieves the balance that’s so necessary,” Mr. Carney said. “And the balance is important because a plan that does not have it puts, unduly, the burden on senior citizens — or on middle-class Americans or on parents with disabled children.”

Mr. Carney also said Mr. Obama is willing to make “tough choices” on spending cuts.

“Any potential agreement would not only have to align with the president’s principles, it would require tough choices by both sides, including the president and including Republicans, and that’s the only way it can come together,” he said.