The Senate’s “gang of six” appears to be headed toward a budget compromise that would boost tax revenues and rein in popular entitlement programs, according to a Democrat and a Republican from the bipartisan group who appeared Sunday on NBC’s “Meet the Press.”

“If we don’t have an agreement soon, we won’t be relevant to this discussion,” Senate Budget Chairman Kent Conrad (D-N.D.) said. “We intend to be relevant. We have made enormous progress in this group. It is the only bipartisan effort that is under way, and at the end of the day it has to be bipartisan or nothing is going to happen.”

The group’s recognition that compromise is required perhaps explains why Conrad is embracing plans to overhaul Medicare, Medicaid and Social Security. Reforming entitlement programs has largely been a priority for Republicans.

It may also explain the willingness of Sen. Tom Coburn to raise additional tax revenues, partially by closing perceived loopholes, despite a pledge he signed with the conservative interest group Americans for Tax Reform.

“Which pledge is most important — the pledge to hold your oath to the Constitution of the United States? Or a pledge from a special interest group who claims to speak for all of America’s conservatives, when in fact they really don’t?” the Oklahoma Republican asked on “Meet the Press.” “We’re not talking about raising tax rates at all. If there’s a net effect of tax revenue, that would be fine with me. I experienced that during Reagan’s period in 1986.”

Coburn added that a compromise that can earn 60 votes in the Senate and a majority in the House is critical.

“The president doesn’t have a plan that’ll get 60 votes,” he said. “The House doesn’t have a plan that’ll get 60 votes. And what Sen. Conrad and myself and other colleagues are trying to do is — where is the compromise that’ll save our country?”

Both men suggested they would not support a long-term plan to raise the nation’s debt limit unless it includes major plans for debt reduction.

“Our country is headed for a fiscal cliff,” Conrad said. “This is a defining moment. We’ve got to decide as a nation, are we going to do some things that all of us would prefer not to have to do, or do we wait for the roof to cave in?”

Sen. Richard Blumenthal, who appeared Sunday on CBS’ “Face the Nation,” said he believes “unequivocally” that Congress will vote to raise the debt ceiling. The spending reductions are the undecided issue, the Connecticut Democrat added.

“I think the only question is really, whether raising the debt ceiling will be tied to spending reductions,” he said. “And I personally believe that we need to rein in government spending, cut the debt and the deficit, but at the same time make our priority jobs and economic growth.”

Sen. Mark Kirk said on “Face the Nation” that raising the debt limit isn’t a certainty.

“Maybe or maybe not,” the Illinois Republican said. “I will vote ‘no’ on raising the debt ceiling unless we have comprehensive, dramatic, effective and broad-based cuts to federal spending, including the reform of entitlement spending.”

Rep. Joe Walsh accused the administration of overstating the consequences of reaching the debt ceiling.

“The administration is playing politics with this issue, just like they’re playing politics with entitlement reform,” the Illinois Republican said in a separate appearance on “Face the Nation.” “What’s going to happen — all we’ve got to do is look in the last 20 years. Three or four times over the course of the last 20 years, Congress has voted not to raise the debt ceiling, and it’s taken a few months and then they’ve come together and they’ve raised it. But over the course of those few months when the debt ceiling wasn’t raised, Armageddon didn’t hit. The government paid its bills. We’ve got enough government revenues to certainly pay — to service all of our debt. And the administration knows that.”

Treasury Secretary Timothy Geithner has stressed that the risk would be catastrophic, but he has expressed confidence that Congress would act in time. The Treasury Department estimates that the nation will hit the ceiling by May 16 and that emergency measures can delay a default until early July.