Republicans and Democrats on Capitol Hill are drafting radically different budget blueprints that offer little room for compromise, even as President Obama presses lawmakers to take another shot at a far-reaching agreement to tame the national debt.

On Tuesday, House Budget Committee Chairman Paul Ryan (R-Wis.) rolled out a 10-year spending plan that would revive the most controversial prescriptions from last year’s GOP budget, including a partial privatization of Medicare and a repeal of the health-care law that is Obama’s signature policy achievement.

Meanwhile, Senate Budget Committee Chairman Patty Murray (D-Wash.) briefed her colleagues on a competing plan, to be released Wednesday, that would raise taxes by nearly $1 trillion over the next decade and spend nearly $100 billion on a new jobs package — ideas Republicans have firmly rejected.

“They’re opening bids. But they’re opening bids from three years ago,” said Robert Bixby, executive director of the bipartisan Concord Coalition, which champions deficit reduction. “The real question is: Do they start a negotiation this year? It’s not where they start, it’s where they finish. So you can take both of these budgets with a big grain of salt.”

Obama seemed to do that Tuesday in a lunchtime meeting with Senate Democrats, the first of four sessions he plans to hold this week with rank-and-file lawmakers in both parties.

While the White House issued a statement criticizing Ryan’s blueprint as “the wrong course for America,” Obama told Senate Democrats to expect a months-long debate over fiscal issues that will begin in earnest only after each chamber has approved its own partisan vision for improving the economy and shrinking the national debt.

“The best course now is to let the budgets go, get them into [a] conference [committee] and try to reconcile the two,” Sen. Benjamin L. Cardin (D-Md.) said the president advised senators during the closed-door meeting.

Still, a senior administration official acknowledged that the dueling blueprints illustrate the immense challenge of trying to forge a compromise between a president and Democratic lawmakers, who insist on a big dose of new tax revenue to reduce borrowing, and Republicans, who refuse to consider any additional revenue beyond the relatively modest tax increase adopted Jan. 1.

Ryan, who campaigned against Obama last year as the GOP vice-presidential nominee, is offering the more uncompromising spending plan by far, one that Democrats say ignores Obama’s convincing reelection victory less than five months ago.

In addition to repealing the president’s health-care expansion, the 91-page blueprint proposes rolling back the administration’s Wall Street reforms and opening federal land to oil drilling. Ryan also would protect the Pentagon from automatic spending cuts known as the sequester by shifting those reductions to domestic agencies. And he proposes to trim domestic agencies by an additional $250 billion over the next decade.

All told, Ryan would slice $4.6 trillion from projected spending, with more than half of those savings — $2.7 trillion — coming from the big health-care programs, primarily Medicaid and Obama’s Affordable Care Act.

That would let him wipe out deficits by 2023 without raising taxes. Ryan also calls for an overhaul of the tax code, but he would eliminate an array of tax breaks to finance a reduction in the top rate to 25 percent from 39.6 percent, a goal Democrats say would reduce taxes for the wealthy at the expense of the middle class.

At a Capitol Hill news conference, Ryan defended his decision to reprise policies that Obama and other Democrats opposed during the 2012 campaign.

“The election didn’t go our way. Believe me, I know what that feels like. That means we surrender our principles?” Ryan said. “We think we owe the country a balanced budget. We think we owe the country solutions to the big problems that are plaguing our nation: a debt crisis on the horizon. A slow-growing economy. People trapped in poverty. We’re showing our answers.”

As Ryan prepared for a vote on his budget in committee this week and before the full House next week, Senate Democrats were crafting a more modest blueprint that nonetheless retreats from the terms of the budget deal Obama has offered House Speaker John A. Boehner (R-Ohio) and, more recently, Republican senators gathered for dinner with the president last week at a Washington hotel.

Full details of the plan will be available Wednesday. Democratic aides and lawmakers briefed on the document said it proposes to replace the sequester cuts with $1.85 trillion in alternative policies over the next decade, including nearly $1 trillion in new taxes — far more than the $600 billion Obama is seeking.

Murray also proposes nearly $1 trillion in spending cuts, with just $275 billion coming from health programs, short of Obama’s $400 billion offer. And she seeks $100 billion for a new “economic recovery protection plan” that would fund infrastructure projects and education programs — and was quickly derided by Republicans as a new burst of “stimulus” spending.

In a statement, Murray called her plan “a responsible pro-growth budget that reflects the values and priorities of the middle class.” But Republicans pounced on the call for new taxes.

After 1,413 days without a spending plan, “all the Democrats can come up with for a budget is a trillion-dollar increase in taxes,” said Sen. Jerry Moran (Kan.).

Despite the sniping, there were signs that both sides are open to a compromise. In contrast to the combative tone Ryan adopted last spring, when a Republican takeover of the White House and the Senate seemed like a real possibility, Ryan this year acknowledged that Obama is unlikely to adopt his proposals. And he repeatedly referred to his spending plan as “an invitation” to open negotiations.

Meanwhile, Senate Minority Leader Mitch McConnell (Ky.) welcomed Obama’s outreach to the GOP, which will continue Wednesday when the president returns to Capitol Hill to meet with House Republicans.

“I told the president on Friday I hope he’ll invite all of our members down for these dinners,” McConnell told reporters. “We all know that with the president’s request to raise the debt ceiling here again, later this summer, we’ll be discussing again the possibility of finally solving our huge deficit and debt problem.”

Paul Kane contributed to this report.