Chambliss, a veteran senator from Georgia, said he signed the pledge during an earlier campaign when the country's debt was nowhere near its current $16 trillion level.
"Times have changed significantly, and I care more about my country than I do about a 20-year-old pledge," Chambliss told his local television station. "If we do it (Norquist's) way, then we'll continue in debt."
"I'm frankly not concerned about the Norquist pledge," Chambliss added.
Raising taxes, whether by closing loopholes or raising tax rates, is seldom a vote-winning strategy.
President George H.W. Bush broke his campaign promise to not raise taxes; he ended up losing re-election in 1992.
Other Republicans, however, are now willing to put additional tax revenues on the table as a bargaining chip for a deal with Democrats to get changes in Social Security and Medicare and pare down federal deficits.
"I agree with Grover, we shouldn't raise rates. But I think Grover is wrong when it comes to we can't cap deductions and buy down debt," Sen. Lindsey Graham of South Carolina said Sunday on ABC's "This Week."
"I will violate the pledge — long story short — for the good of the country, only if Democrats will do entitlement reform," he added.
Rep. Peter King of New York told NBC's "Meet the Press" on Sunday that the pledge is good for a two-year term only.
"A pledge you signed 20 years ago, 18 years ago, is for that Congress," King said. "For instance, if I were in Congress in 1941, I would have signed a support of declaration of war against Japan. I'm not going to attack Japan today. The world has changed, and the economic situation is different."
Sen. John McCain, the Republican presidential nominee in 2008, said the pledge is losing its clout.
"Fewer and fewer people are signing this, quote, pledge," he told an audience recently.
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