How big is the nation’s borrowing problem? Is it the $6.3 trillion in fresh debt projected over the next decade? Or is it the more than $70 trillion that Republicans say the nation will rack up over the next 30 years?

Under Option 1, the national debt is high but stable as a share of the nation’s overall economy. Under Option 2, the debt is exploding to levels unseen in U.S. history.

The difference helps explain one of the most fundamental disagreements between the political parties as budget talks lurch into their third-straight year on Capitol Hill: Democrats, looking at the short term, have declared the nation’s debt problem all but solved. Republicans, looking at the long term, are demanding huge additional cuts in spending.

In recent days, Republicans have sought to sharpen their point in the dispute, arguing in meetings with top White House officials that the 30-year “window” offers a clearer view of the challenges ahead.

“You need to attack the 30-year problem,” said Sen. Ronald H. Johnson (Wis.), one of more than a dozen GOP senators involved in the meetings. “The budget problem we’re facing is really about the demographic one, spanning the baby boom generation.

The battle over the 2014 budget

He added: “As we move into retirement until, I hate to say it, the end of our lives, we’ve got promises made to that generation that are going to be very difficult for this government to keep. Unless you’re willing to look at that honestly, unless you’re willing to tell the American people the truth, I don’t see how you can start solving the problem.”

Democrats are practically rolling their eyes, dismissing the argument as yet another stall tactic to avoid addressing the most immediate budget issue: what to do about the sharp automatic cuts known as the sequester.

“I’m happy to talk about Medicare 30 years down the road,” said Senate Budget Committee Chairman Patty Murray (D-Wash.). “But we can’t put all the what-ifs on the table right now and say we’re not going to take care of the horrific impacts of sequestration because we’re busy figuring out everything that’s going to happen in the next three decades.”

The sequester is slicing roughly $40 billion out of agency spending this year. That figure will more than double in the fiscal year that begins in October, slowing economic growth in the coming calendar year by 0.7 percentage points, according to independent estimates.

Republicans want to keep the sequester cuts but exempt the Pentagon. President Obama wants to replace most of the cuts for all agencies with higher taxes on the rich and long-term reforms to Social Security and Medicare that would save $1.8 trillion over the next decade and, some Republicans acknowledge, even more over the next 30 years.

But other Republicans have rejected Obama’s approach because it involves higher taxes. Unless the two sides can reach an agreement, the government will shut down Oct. 1, and the Treasury Department will face a potential default before the year is out.

During an hour-long session Tuesday afternoon in a Senate hearing room, White House officials were more receptive to the GOP argument, according to Johnson and other Republicans present, agreeing to use the long view to examine potential policy options.

The meeting was the second in less than a week between Republican senators and White House chief of staff Denis McDonough. Deputy chief of staff Rob Nabors and budget director Sylvia Matthews Burwell also attended. Afterward, McDonough would say only that it was “a good set of conversations.”

Republicans said the talks remained stuck on the question of what the GOP is willing to offer in place of Obama’s plan. Johnson criticized Obama’s proposal to apply a less-generous measure of inflation to Social Security cost-of-living increases, for example, noting that it would solve barely 20 percent of the program’s long-term cash shortfall. But Republicans have offered no proposals for shoring up Social Security’s finances.

Some were even leery of Johnson’s plan to use a 30-year budget window. Such a frame, they said, could encourage a compromise that would trim today’s sequester cuts in exchange for future reductions in Social Security and Medicare — a deal that would amount to trading concrete cuts for wispy promises of restraint.

What did seem clear after Tuesday’s meeting, several Republicans said, is the need to winnow the group of senators who have attended dinners, golf outings and other meetings with Obama — jokingly dubbed “the diners’ club” by GOP aides. Instead, they said, four or five senators should be authorized to begin negotiating in earnest.

Until that happens, Sen. Saxby Chambliss (R-Ga.) told reporters, there’s “no negotiations,” just “conversations.”

“We are not yet at the point,” added Sen. Rob Portman (R-Ohio), “where we’re ready to talk about solutions.”