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“We’re not going to balance the budget in 10 years” in the manner Mr. Ryan proposes, Mr. Obama told ABC News. “My goal is not to chase a balanced budget just for the sake of balance. My goal is how do we grow the economy, put people back to work? And if we do that, we are going to be bringing in more revenue.”

Mrs. Murray’s proposal and any likely Obama budget are dead on arrival in the Republican-controlled House, while the Ryan blueprint will die in the Democrat-run Senate.

But it’s not a given that the Murray plan will even survive her own chamber, as several moderates in her party are up for re-election next year in conservative states — including Mark L. Pryor of Arkansas, Mark Begich of Alaska, Kay R. Hagan of North Carolina, Tim Johnson of South Dakota and Mary L. Landrieu of Louisiana — and thus may find it difficult to support the plan’s tax increases.

Mr. Begich, when asked whether he could support a budget plan that doesn’t balance in a time-certain fashion, suggested the Murray blueprint could be tweaked.

“We’re looking to balance it — I don’t worry about those things,” he told reporters.

Budget plans are nonbinding, but they set the tone for the legislative initiatives and debates for each party and chamber for the rest of the year. Some provisions in the blueprints likely will be spun off into separate bills.

Senate Republicans — who have mocked Senate Democrats for failing to submit a budget — haven’t offered a plan of their own.

From the Senate floor, Majority Leader Harry Reid, Nevada Democrat, said he expected a “kinder, gentler” Republican Party after it lost the presidential race and congressional seats in the November elections, only to find a Ryan budget that is “anything but balanced.”

Mr. Ryan acknowledged his party’s setbacks at the ballot box, but said it would not surrender its principles. The attempts to eliminate Mr. Obama’s health care law, however, are effectively futile with a Democratic majority in the Senate and Mr. Obama’s veto pen.

Mr. Ryan’s plan, titled “The Path to Prosperity,” would simplify the tax code into two brackets, of 10 percent and 25 percent, and repeal a Wall Street regulatory bill known as Dodd-Frank and the high-speed rail programs promoted by Vice President Joseph R. Biden.

On Medicare reform, Mr. Ryan asked Congress to keep benefits in place for those near retirement, yet transform the program for younger Americans by allowing private plans to compete with the traditional fee-for-service model. The competition would take place within a “Medicare Exchange,” a concept that evokes the marketplaces under Mr. Obama’s law.

Rep. Chris Van Hollen, Maryland Democrat and ranking member of the House Budget Committee, criticized his Republican counterpart for crafting a plan that accounts for revenue from the health care law while trying to repeal it, and for retaining cuts to Medicare that Mr. Ryan blasted on the campaign trail last year.

“You just can’t have it both ways,” the congressman from Maryland told reporters.

Sniping over the budget plan arrives days after Mr. Obama sat down with Mr. Ryan and Mr. Van Hollen, as some on Capitol Hill hold out hope for a “grand bargain” to rein in the nation’s debt.

“I don’t think it means we can’t get there,” Mr. Van Hollen said of the discord. “It shows the gulf we have to bridge is wide.”