Speaker John Boehner has offered House Republicans a choice in response to GOP members grumbling about the Senate “fiscal cliff” deal that passed at 2 a.m. New Year’s Day, according to a House leadership aide.

“The Speaker presented his members two options,” the aide explained in an email. “The first would be to make an amendment to the Senate bill that would add a package of spending cuts. The Whip will do a whip check on this spending cuts amendment after the meeting. If we can get the commitment of 218 votes on this amendment, we will bring it to the floor and send it to the Senate. The Speaker and the Leader both cautioned members about the risk in such a strategy. They told them there is no guarantee the Senate would act on it. If we cannot get the commitment of 218 votes tonight, we will bring up the Senate-passed measure for an up-or-down vote in the House.”

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The big question now is whether House Republicans, who scoffed at Boehner’s proposal to raise rates on income above $1 million, would accept the Senate bill’s $400,000 cut off for individuals and $450,000 for families in exchange for more spending cuts. If not, then the Senate bill could conceivably pass as is with a lot of Democratic votes.