Facing the so-called fiscal cliff of automatic tax hikes and deep across-the-board spending cuts at the end of the year, the GOP legislators are signaling their willingness to cut a deal with President Barack Obama and Democrats that would include more money for the government. Poll: Lawmakers won't act their age on fiscal cliff The overall numbers remain relatively small -- a handful of senators and House members -- but they include influential veterans such as Sens. Saxby Chambliss of Georgia and Lindsey Graham of South Carolina, along with Rep. Peter King of New York. Others who have rejected the strict dogma of the Norquist pledge include Sens. John McCain of Arizona, Tom Coburn of Oklahoma and Bob Corker of Tennessee, as well as Rep. Scott Rigell of Virginia, who was elected in the tea party wave of 2010 and recently re-elected. Norquist, who founded the conservative Americans for Tax Reform, advocates shrinking government by cutting spending instead of raising taxes through higher rates or reforms. Norquist: I'll unseat pledge-breakers At the White House, spokesman Jay Carney told reporters he hoped the "welcome" comments by some Republicans represented "a difference in tone and approach to these problems." To CNN Chief Political Analyst Gloria Borger, the softening tone by some in the GOP was explained by new poll numbers that showed 45% of Americans would blame Republicans for failing to avoid the fiscal cliff while 34% would blame Obama. The public shift comes after Obama won re-election and Democrats increased their slim Senate majority and narrowed the GOP majority in the House in this month's election. In what have been secret talks so far, Obama and Congress are seeking to revive a possible "grand bargain" to cut the chronic federal deficits and debt. In addition, spending cuts would reduce spending on the military, national parks, the FAA and other important government services. However, the government and Congress still would have time to prevent draconian impacts from the fiscal cliff when a new Congress convenes in January. William Galston, a senior fellow in governance studies at the Brookings Institution, called that a form of brinksmanship best avoided. To be sure, no one believes that non-agreement by December 31 would be the end of the story. After a period of finger-pointing, discussions would resume," Galston wrote last week in a New Republic opinion piece. But equally, no one knows how the failure to reach agreement before the end of 2012 would affect the dynamics of the negotiations. In addition, "we can be reasonably sure ... that national and global markets would react adversely and that businesses, which are already retreating from planned investments in new plant and equipment, would become even more uncertain and risk-averse." The CNN/ORC International poll released Monday also showed that a solid majority of respondents -- two-thirds -- supports the Democratic stance that any agreement should include a mix of spending cuts and tax increases. Come January, there will be 39 senators, including Chambliss and Graham, and 219 House members who endorsed it. Trending: Graham says he'd break no-tax 'pledge' The House total constitutes a narrow majority in the 435-seat chamber, though some members have denounced their allegiance to the pledge. Some congressional conservatives sought to deflect attention from the Norquist pledge on Monday, focusing instead on the need to work out a deal that included concessions by Democrats. "The goal is to solve the problem," insisted Rep. Kevin McCarthy, the No. 3 House Republican in the incoming Congress. He rejected Obama's call for letting tax rates on income over $250,000 return to higher levels of the 1990s, telling CNN "that doesn't solve the problem" because "you do nothing about the growth of government." Boehner, the Ohio Republican who has emerged as party leader in the deficit talks, agrees to the concept of increased revenue, though he and Senate Minority Leader Mitch McConnell of Kentucky both remain opposed to actually raising tax rates. Fiscal cliff tax deal: Getting to $1 trillion Read more: 8 things that could kill a fiscal cliff deal Opinion: Can GOP handle prosperity? CNN's Paul Steinhauser, Thom Patterson and Kevin Liptak contributed to this report.