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."
Long a defining difference between Democrats and Republicans, the tax issue has stymied efforts to work out a deficit deal for the past two years.
Obama and House Speaker John Boehner came close to agreement last year before conservative rejection of any increased revenue and liberal resistance to entitlement reform scuttled the effort.
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.
Instead, they propose broad tax reform that will lower rates while eliminating unspecified loopholes and exemptions to spur economic growth that they say will result in more overall government revenue.
"It's fair to ask my party to put revenue on the table. We're below historic averages," Graham told ABC. "I will not raise tax rates to do it. I will cap deductions. If you cap deductions around the $30,000, $40,000 range, you can raise $1 trillion in revenue, and the people who lose their deductions are the upper-income Americans."
At the same time, Graham and other conservative lawmakers demand that Democrats agree to significant reforms in entitlement programs such as Medicare and Medicaid, the government-run health care programs for senior citizens, the disabled and the poor.
"I will violate the pledge, long story short, for the good of the country, only if Democrats will do entitlement reform," Graham said.
On the same program, Democratic Sen. Dick Durbin of Illinois said some changes to Medicare are needed, but he ruled out any reforms to Social Security, the national retirement plan, saying it is a separately funded system that "does not add a penny to our debt."
Noting opposition to entitlement reforms by traditional Democratic allies such as organized labor, Durbin said everyone has to realize that some changes are needed in Medicare and Medicaid.
"Those who say 'don't touch it, don't change it' are ignoring the obvious," said the Senate's No. 2 Democrat, adding that "we can make meaningful reforms in Medicare and Medicaid without compromising the integrity of the program, making sure that the beneficiaries are not paying the price for it, except perhaps the high-income beneficiaries. That to me is a reasonable approach."
However, Durbin balked at one proposal sought by Republicans -- to slowly raise the eligibility age for Medicare above the current level of 65.
"What happens to the early retiree who needs health insurance before that person's eligible for Medicare?" Durbin asked. "My concern about raising that Medicare retirement age is there will be gaps in coverage or coverage that's way too expensive for seniors to purchase."
Graham rejected Durbin's point, saying the same change instituted in Social Security has worked. He also called for adjusting benefits based on the personal wealth of recipients, so that those with more money have to pay more for services.
Norquist said Democrats will never agree to the negotiating position of Graham and other Republicans, calling the demand for entitlement reforms akin to a "pink unicorn that doesn't exist."
The political risk for Republicans to going against the no-tax pledge comes from angering Norquist and other conservatives who can target them in GOP primary campaigns in 2014 and beyond.
Norquist said Monday his group would "certainly highlight who has kept their commitment and who hasn't" when re-election time comes.
"The key here is whether or not the Republicans will move away from the ideologically rigid position, which has been the Grover Norquist pledge," Democratic Sen. Carl Levin of Michigan told NBC on Sunday. "You've got to raise additional revenues, including tax rates on the wealthy. They have to go up. Either real tax rates or effective tax rates, there are ways of doing that."