Meanwhile, conservative interest groups are trying to pressure Republicans to oppose any tax boosts.
"If they vote for tax rate increases, they're likely going to have a primary opponent and have a tough 2014," Chris Chocola, president of the conservative Club for Growth, said in an interview.
The group spent $10 million in the 2012 campaigns opposing GOP candidates it considered not conservative enough, according to the nonpartisan Center for Responsive Politics, which tracks election spending.
"I would presume a Republican primary would be a red flashing light in many places," said Rep. Tim Huelskamp, R-Kan., another freshman disciplined by Boehner who says he will oppose tax increases.
Americans for Tax Reform, headed by anti-tax activist Grover Norquist, has sent a petition to the 150,000 people on its email list.
Most GOP lawmakers have signed Norquist's pledge promising to oppose tax increases, though some now say addressing the government's fiscal problems outweighs the pledge.
The petition urges lawmakers "to keep the promise they made to their constituents to not raise taxes." More than 38,000 people had signed it by late last week, said group spokesman John Kartch.
Associated Press writer Philip Elliott contributed to this report.
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