Senator Lindsey Graham has become the second senior Republican in days to publicly disavow a pledge that handcuffs the party to a policy of no tax rises, raising hopes of a deal over the fiscal cliff.

Speaking on ABC's This Week, the South Carolina politician said that the only pledge members of either party should make would be one to make sure the country did not go the same way, economically, as Greece. Regarding a pledge against tax hikes that has been signed by most Republicans in Congress – having been promulgated by the conservative lobbyist Grover Norquist – Graham said: "I will violate this pledge, long story short, for the good of the country."

In the aftermath of the GOP's defeat in the presidential election of 6 November, Norquist is increasingly seeing his influence on the party decline. On Thursday, senator Saxby Chambliss said he would break the Taxpayer Protection Pledge in an attempt to help avert the automatic triggering of $600bn of spending cuts and tax increases, the so-called "fiscal cliff".

Washington has until the end of a year to negotiate a deal to avoid such a situation. Economists have suggested that a package of swingeing spending cuts and tax hikes could be catastrophic for the US economic revival, plunging the country back into recession.

"I care more about this country than I do about a 20-year-old pledge," Chambliss said in an interview, earning a rebuke from Norquist.

A vast majority of Republicans in the House and the Senate have signed the Taxpayer Protection Pledge, which was created in 1986 and which commits them to voting against any increase in revenue taken from personal income. Until recently it had been seen as a litmus test for the conservative credentials of party representatives, but the incoming House of Representatives will have 16 Republicans who have not signed up – an increase from six. One new Republican senator, Arizona's Jeff Flake, has avoided putting his signature to the demand.

On Sunday, Graham – who had already spoken of his misgivings about the pledge – added his name to those who have gone on record about their intention to break with the policy.

"I think Grover is wrong," he said. "When you are $16tn in debt the only pledge we should be making to each other is to avoid becoming Greece. I will violate this pledge, long story short, for the greater good of the country, only if Democrats do entitlement reform."

Speaking on the same show, the Democrat senator Dick Durbin also indicated a willingness for negotiation. After saluting Graham, Durbin said: "What he just said about revenue and taxes needs to be said on his side of the aisle. We need to be honest on our side of the aisle."

Durbin, a Democratic Party whip, noted that Congress was due to begin its new session on Monday.

"We can solve this problem," he said. "Tomorrow, there's no excuse: we're back in town."