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Republicans were still tweaking the final version of their bill late Wednesday in order to try to win enough support.

Mr. Boehner says his plan will keep taxes low for 99.8 percent of taxpayers, and would amount to a $3.9 billion tax cut compared with the current law.

The White House, though, said Mr. Boehner’s Plan B would amount to a $50,000 tax cut for millionaires, who will still pay lower rates on their income up to $1 million.

White House Communications Director Dan Pfeiffer faulted Mr. Boehner’s plan for what it left out: It does not extend unemployment benefits, it does not end the looming $110 billion in spending cuts, and it fails to boost payments for doctors who treat Medicare patients.

Republicans said their Plan B was not intended to solve the entire fiscal cliff, but rather to be a short-term solution to make sure tax rates didn’t rise for most Americans on Jan. 1.

Seth McLaughlin and Susan Crabtree contributed to this report.