With the prospect of a government shutdown looming Friday, leaders of both parties publicly staked out seemingly inflexible positions while staff members worked in private on a possible compromise to finally pass the 2011 budget.

If they come up with a deal that both sides can accept, the agreement could become a blueprint for other even more contentious budget battles on the horizon. If they fail, then the government will be shut down as happened in 1995 and 1996 — with inevitable disruptions and unpredictable economic and political consequences.

Both Democratic and Republican aides said efforts continued over the weekend to fashion a bill with $33 billion in spending cuts, but they also said neither side intends to officially announce a “deal” on that plan.

Instead, once staff members from the House and Senate appropriations committees finish their effort, the legislation will be presented to rank-and-file lawmakers to determine if there are enough votes to pass it. The wild card remains the 87 House Republican freshmen who won in the fall largely on tea party pledges to slash government spending and who are pushing for much deeper cuts.

In addition to the political and policy obstacles to overcome, the legislators also face new procedural hurdles that could make a deal more difficult to achieve on time.

Under new House Republican rules, any bill to be voted on Friday would have to be posted by Tuesday night. Republican leaders have also come out against approving another stopgap measure to keep federal agencies open a few extra days as they finish their work.

For now, Democrats and Republicans are operating in a parallel legislative universe, at least publicly.

Led by Vice President Biden, Democrats have embraced the effort to find $33 billion in cuts compared with last year’s funding levels, while House Speaker John A. Boehner (R-Ohio) has publicly denied agreeing to that number and has said he is pushing for deeper spending cuts.

At the same time, senior aides and lawmakers acknowledge that $33 billion is the goal, with the key hurdles being which items would be cut and whether Republicans can win the inclusion of policy prescriptions on social and regulatory issues.

Democrats are also pushing to draw savings from programs that are outside the annual appropriations bills that fund most of the federal government, contending billions of dollars in cuts could come from mandatory spending programs inside the Agriculture and Treasury departments.

The time crunch has left some lawmakers bracing for a shutdown. Rep. Chris Van Hollen (D-Md.), whose suburban Maryland district is home to 80,000 federal workers, said he thinks there is at least a “50-50 chance” of a shutdown. The top Democrat on the House Budget Committee, Van Hollen said that even a government closure for several days would send a “a very bad signal” to American voters and the international community that Washington cannot function.

“I think it harms people’s confidence in the federal government,” Van Hollen said.

House Republicans, who six weeks ago approved a plan to slash $61 billion from federal agencies, have accused President Obama of being too distant and hands-off with the budget impasse.

“We can’t keep kicking this can down the road. The president has punted. We’re not going to follow suit,” House Budget Committee Chairman Paul Ryan (R-Wis.) said on “Fox News Sunday.”

Senate Majority Leader Harry M. Reid (D-Nev.) saw things differently. Speaking on CBS’s “Face The Nation,” he said the fate of negotiations rests on whether Boehner stops tea party activists’ “dictating” to leadership on what sort of deal he can accept.

“It’s so easy to do,” Reid said. “It’s just really, in Washington terms, a few dollars short of being able to do this. It’s a question of how we do it.”

Today’s dispute — a debate over the last six months of fiscal 2011 funding — is small compared with the budget and debt debates awaiting Congress.

On Tuesday, Ryan intends to unveil his much-anticipated budget proposal for fiscal 2012. Over 10 years, he said Sunday, his plan would carve out more than $4 trillion in savings, much of it coming from a dramatic reshaping of health programs for the poor and elderly. Medicaid, under Ryan’s proposal, would be turned into a block grant program in which the federal government would ship money to states, where governors would have the “freedom” to implement the plans as they see fit.

Some Republicans think Ryan’s 2012 budget proposal will serve as a catalyst to rounding up enough votes for the 2011 spending plan, but first they have to get the current-year plan to the House floor.

Under new rules, Republicans declared that legislation must be public for 72 hours before it can be considered on the floor. That means they would need to have a 2011 spending bill completely written and posted online by Tuesday night to vote Friday night, something aides say is possible but not definite. That leaves almost no time for the Senate to act — and conservatives there would have many parliamentary tactics to extend the debate for several days past the Friday deadline.

Some House aides have floated the idea of breaking the 72-hour pledge, but such a move would be risky, given that the rule was an offering to the tea party activists who accused Democrats of crafting deals behind closed doors and rushing them to the floor when they ran the House.

The only option to avert a brief shutdown might be the passage of a seventh short-term funding plan to be approved, but GOP leaders have said publicly they do not want to continue funding the government with these bite-size measures.

Boehner’s leadership team has tried to convince many Republicans that once they move beyond this year’s budget fight — over a sliver of the overall government spending — they can move into Ryan’s much broader budgetary overhaul.

But some Republicans have said compromising on the smaller issue now will only set themselves up for failure on the bigger issue of the 2012 budget and a vote on whether to raise the federal debt limit later this spring.

“How are we ever going to deal with the bigger issue of the national debt if we can’t cut $60 billion now?” freshman Rep. Tim Griffin (R-Ark.) said at a news conference on the Senate steps Friday. “If we can’t trim pennies now, how are we going to deal with dollars in a month or so on the broader issue?”