President Obama and House Speaker John A. Boehner escalated their battle over the national debt on Monday, pressing their arguments in a pair of prime-time television addresses as Congress remained at a loss over how to keep the United States from defaulting next week for the first time.

The challenge facing any plan for reducing the debt was underscored when a new Republican proposal to raise the ceiling on federal borrowing was met Monday with misgivings by some conservatives and skepticism by many GOP freshmen. That called into question whether Boehner (R-Ohio) could even get his own caucus to back his approach.

As Boehner tried to rally support for his two-step plan to cut $3 trillion in spending, Senate Majority Leader Harry M. Reid (D-Nev.) offered a strikingly similar proposal for increasing the debt limit before the Aug. 2 deadline. The two leaders, however, remained bitterly divided over Boehner’s demand to hold another vote next year to further expand the government’s borrowing authority.

With financial markets warily watching the Capitol Hill drama, Obama used his 15-minute address from the White House to urge “shared sacrifice” in tackling the debt, calling for deep cuts in federal spending to be coupled with higher taxes on the wealthy and on large corporations.

He slammed Boehner for calling for another vote on the issue next year, saying: “We know what we have to do to reduce our deficits; there’s no point in putting the economy at risk by kicking the can further down the road.”

Boehner countered with a shorter speech from the Capitol, in which he blamed the fiscal crisis on Washington’s spending and urged deep cuts to cure it. He said Reid’s plan lacks the kind of real spending cuts needed for the government to operate within its means.

“While the Senate is struggling to pass a bill filled with phony accounting and Washington gimmicks, we will pass another bill,” Boehner said, predicting his approach would prevail.

The government has exceeded its $14.3 trillion debt limit, and Treasury Secretary Timothy F. Geithner has said that without action by Aug. 2, the government will not be able to pay its bills. Credit-rating firms warned that they could downgrade the U.S. debt, which could spur higher interest rates and cause aftershocks in global markets.

Absent an agreement between Boehner and Reid, the House and the Senate are headed for a high-wire act this week.

Neither leader was certain that he could rally the votes to win — with Boehner making the first move for a possible Wednesday vote. With few House Democrats expected to support his approach, Boehner would need the support of an overwhelming majority of his 240-member conference.

But those hopes were dampened Monday by conservative opposition to the plan, highlighted by Rep. Jim Jordan (Ohio), who leads a conservative caucus of more than 170 GOP members. Jordan is one of 39 House Republicans who previously took a pledge vowing to increase the debt ceiling only in return for Congress sending to the states a constitutional amendment requiring a balanced budget.

Reid’s measure faces its own hurdles because Republicans criticize some of his savings as accounting tricks. So Reid continued private talks with Senate Republicans in an effort to modify Boehner’s package to make it more palatable to Obama, who has previously said he would veto any proposal that provided only a short-term increase in the debt ceiling.

Both plans call for deep reductions to federal agencies over the next decade, and neither would immediately require increased tax revenue or deep cuts to entitlement programs, such as Medicare.

Boehner’s plan includes an immediate increase in the debt ceiling of up to $1 trillion, paired with $1.2 trillion in spending cuts over the next decade. A new bipartisan committee of 12 lawmakers would be directed to come up with an additional $1.8 trillion in deficit savings and send its recommendations to Congress for fast-track approval without any amendments allowed.

If such a plan were approved, Obama could request up to $1.5 trillion in new borrowing authority early next year, and Congress could block it only with a two-thirds majority in each chamber.

The Reid bill includes upfront spending cuts that would exceed the size of the proposed increase in the debt ceiling. This approach, too, would create a 12-member committee to produce a broad strategy for reducing the deficit by the end of the year.

Although both bills propose to decrease agency spending, including $1.2 trillion at the Pentagon over the next decade, congressional budget analysts would credit the Reid bill with an additional $1.5 trillion in debt reduction.

If the Boehner bill passed the House, Senate Democrats might use the legislation as the basis for raising the debt ceiling. They were continuing Monday to work with Minority Leader Mitch McConnell (R-Ky.) on ways to modify the proposal, Democratic aides said. If the Boehner measure failed, Senate Democrats would push for their own measure.

Highlighting the stakes were the dueling prime-time addresses, which were scheduled only hours beforehand.

Speaking from the East Room, Obama warned that, without compromise, the government “would not have enough money to pay all of our bills — bills that include monthly Social Security checks, [and] veterans’ benefits.”

In the professorial tone that has won admirers and critics, Obama explained what a default and the resulting downgrade of U.S. credit would mean.

“Interest rates would skyrocket on credit cards, on mortgages and on car loans, which amounts to a huge tax hike on the American people. We would risk sparking a deep economic crisis — this one caused almost entirely by Washington,” he said.

He rejected Boehner’s proposal for a short-term increase in the debt ceiling because it would leave the economy “held captive” to the threat of a default. “This is no way to run the greatest country on Earth. It’s a dangerous game that we’ve never played before, and we can’t afford to play it now,” the president said.

In remarks that immediately followed Obama’s, Boehner said he “gave it my all” in negotiating with the White House over a “grand bargain” that could have yielded trillions of dollars in savings and raised the debt ceiling through 2013. That effort failed, Boehner said, because Obama had increased his demand for more tax revenue, which Republicans adamantly oppose.

“The president has often said we need a ‘balanced’ approach — which in Washington means: We spend more, you pay more,” Boehner said from his office suites at the Capitol.

The speaker also challenged Obama to help him break the “grip” of the national debt on the stagnant economy. “We are up to the task, and I hope President Obama will join us in this work,” Boehner said.

The speeches ended an intense day in Washington. Anxiety hung over several closed-door party caucuses across the Capitol.

After a 75-minute briefing Monday afternoon, rank-and-file Republicans exited a basement room at the Capitol with mixed reviews of Boehner’s approach. He can count on a core group of at least 100 longtime GOP members, lawmakers said.

Many Republicans, particularly among the 87 freshmen, said they need more time to process the complex details of a proposal that would set up a handful of votes over the next six months.

“I think there are some very interesting parts, and the bill has not been put in print yet, and until it’s put in print, I don’t know that I can say one way or the other,” said Rep. Diane Black (Tenn.), whom GOP leaders often point to as a favorite among the newcomers.

Asked Monday if his plan could win enough Republican votes, Boehner deferred to House Majority Whip Kevin McCarthy (Calif.).

“We ask all Democrats that want to join with us to put this House on the right path, that they could join with us on this bill,” said McCarthy, the party’s top vote counter.

Democrats, in turn, are urging Republicans to back Reid’s approach, saying GOP members have previously supported every item in the package. Among these are $100 billion in savings identified during bipartisan talks led by Vice President Biden, including up to $15 billion in cuts to farm subsidies and $40 billion from reducing fraud and abuse in government programs. There also is $400 billion in savings from reduced interest payments on a lower national debt and $1 trillion from winding down the wars in Iraq and Afghanistan.

House Republicans have been reluctant to count war savings in debt-limit talks, but Democrats note that House Budget Committee Chairman Paul Ryan (R-Wis.) included those savings in the budget blueprint that passed the House this year.

Still, a Ryan spokesman dismissed the Reid proposal.

“An honest budget cannot claim to save taxpayers’ dollars by cutting spending that was not requested and will not be spent,” Conor Sweeney said in an e-mail. “Senate Democrats are employing a budget gimmick that will not fool the credit markets.”

But Reid said his proposal represents the only option for pushing a debt-limit increase through Congress — and getting it signed by the president.

“This isn’t a game of chicken. This is a game of reality,” Reid told reporters. “We’re about to go over a cliff.”