Now that the long-dreaded sequestration cuts have started, the focus in the contentious military-budget-cutting debate shifts to the House as it plans to weigh a plan to make the cuts more manageable for the Pentagon.

President Barack Obama and congressional leaders reached no last-minute deal during a meeting Friday morning to stop the politically unpopular reductions–of $1.2 trillion to decade-long defense and non-defense spending–from starting that same day. Lawmakers emerged from the hour-long White House huddle by basically reiterating their entrenched stances, with Democrats demanding and Republicans rejecting new revenues in any sequestration-replacement plan. Yet they and Obama did offer the Pentagon hope that it will receive some relief from its current budgeting bind.

House Speaker John Boehner (R-Ohio) said the House will debate during the upcoming week a “continuing resolution” (CR) that funds the government past March 27. The measure would give the Pentagon more power to shift around funding within its coffers, House Republicans said.

That new House CR–which would replace another such temporary resolution due to expire–would keep the sequestration cuts but also include a full-blown fiscal year 2013 defense appropriations bill. So, though Pentagon spending is set to be cut by $46 billion for the final seven months of FY ’13 under sequestration, with a complete appropriations bill in place the Pentagon could reprogram funding more easily and lawmakers could help direct where the cuts would be made. House Republicans also want to use that legislation to expand the Pentagon’s authority to shift around funding. Under the current sequestration scenario, the $46 billion in cuts uniformly slice 13 percent off the top of plans, programs, and activity (PPA) accounts from March until October.

Obama told reporters at the White House Friday he is open to signing a new CR that factors in the sequestration cuts–even though he opposes those reductions–because he doesn’t want a government shutdown after the current CR expires March 27. He said he wants the new CR to stick to spending limits in the Budget Control Act of 2011.

“The sequester are additional cuts on top of that,” he acknowledged at a press conference. “And by law, until Congress takes the sequester away, we’d have to abide by those additional cuts.  But there’s no reason why we should have another crisis by shutting the government down in addition to these arbitrary spending cuts.”

House Minority Leader Nancy Pelosi (D-Calif.) told reporters the same day that she agrees with Obama’s views on heeding Budget Control Act spending in a new CR. While she would not comment on the House Republicans’ CR plan, saying she had not seen it, she said she does not want a government shutdown.

Pelosi also spoke in favor of giving the Pentagon more leeway to reprogram funding in FY 13, “so that the harm that can be done to our national security (by sequestration) can be mitigated.”

House Armed Services Committee (HASC) Republicans told reporters at a joint Friday press conference that they see the CR as one several of avenues to addressing sequestration for the military.

“We’re going to come forward with proposals that help mitigate these consequences for national defense” as soon as this upcoming week, said HASC Seapower and Projection Forces subcommittee Chairman Randy Forbes (R-Va.)

“Just because it’s March 1st doesn’t mean we’re folding our hands and saying this is the way it’s got to be,” HASC Vice Chairman Mac Thornberry (R-Texas) said. He noted the upcoming House debate over a CR with a full FY ’13 defense appropriations bill.

“Now that’s not going to undo sequestration, but it is going to add flexibility and it is going to help update the categories, which will reduce some of the damage that comes from having a continuing resolution and a sequestration at the same time,” Thornberry said.

Democrats in control of the Senate, though, have resisted proposals to give the Pentagon more flexibility to manage the sequestration cuts without similarly aiding the other impacted federal agencies.

At the Pentagon, newly sworn-in Defense Secretary Chuck Hagel told reporters Friday afternoon that he met with military leaders about their concerns regarding “uncertainty” caused by both the sequestration and the CR. The Pentagon does not want Congress to simply extend the current CR, which restricts its budgeting flexibility, and hopes it will pass a full FY ’13 defense appropriations bill this month.

Hagel highlighted actions the Pentagon is “taking as a result of these budget constraints,” including the gradual stand down of at least four Navy wings, starting in April. Air Force flying hours will be cut back immediately and the Army will curtail training for all units except those deploying to Afghanistan, he added. The Pentagon plans to issue preliminary notifications later this month to thousands of civilian employees about furloughs. These actions come on top of those the Pentagon took back in January, in advance sequestration’s start, including delaying Navy ship deployment and maintenance plans and also reviewing contracts for delays.

Pentagon program managers are looking at sequestration’s impact on their 2,500 investment programs, Deputy Defense Secretary Ashton Carter added.

“So we’re working with our industry partners on each of those and you’ll see them begin to make adjustments, for example, in the number of weapon systems in a given category that are being purchased,” Carter said at the Pentagon press conference. “So a different kind of arrangement, fewer weapon systems in a contract than we anticipated in a contract. That’s the kind of thing you’ll see.”

Carter said sequestration’s impact “progressively builds over (the) coming months and really constitutes a serious problem, particularly in the readiness accounts.”

Meanwhile, Obama and lawmakers looked forward to the continued debate over a plan to stop or change sequestration.

Boehner stuck to his insistence that he will allow no new taxes, after Republicans already agreed in January to a rate increase on wealthy Americans.

“This discussion about revenue, in my view, is over,” Boehner told reporters on Friday. “It’s about taking on the spending problem here in Washington.”

He called on the Senate to act, noting the House already passed legislation to stop sequestration. Those House GOP plans were non-starters with Democrats because they target spending in areas such as food stamps.

Senate Minority Leader Mitch McConnell (R-Ky.) said in a statement Friday, before the White House meeting, that there would be “absolutely no agreement to increase taxes.”

Obama told reporters he wants Congress “to move forward on a package of entitlement reforms, tax reform–not raising tax rates, identifying programs that don’t work, coming up with a plan that’s comprehensive and that makes sense.”

The president pledged to tackle reforms to Medicare, despite his fellow Democrats’ resistance to entitlement cuts.

“There are members of my party who violently disagree with the notion that we should do anything on Medicare,” Obama said. “And I’m willing to say to them, I disagree with you, because I want to preserve Medicare for the long haul.  And we’re going to have some tough politics within my party to get this done.”

Pelosi said she was open to talking about ways to “strengthen” entitlement programs in a plan to replace sequestration, which she said should include revenue-generating reforms to the tax code.

HASC Chairman Howard “Buck” McKeon (R-Calif.) and the other committee Republicans, for their part, said they will accept no more defense cuts in any alternative plan to sequestration.

“We are telling the president and John Boehner when you walk out of that meeting this morning, don’t plan on cutting our national defense one more cent,” McKeon said Friday morning. Some House GOP members have been more open to allowing defense cuts, as they struggle to secure funding reductions in the acrimonious debate over reducing the deficit.

Pelosi, meanwhile, said even though she wants the Pentagon to have more flexibility to reprogram funding because of sequestration, “that doesn’t mean we don’t think that at some point we should be reducing the defense budget.”

“But that’s more related to what is our mission, what is its cost, how do we prioritize that,” the House Democratic leader said. “This (sequestration) is not the way to cut the defense budget, to have a meat-axe across-the-board cut that harms our national security needlessly and…mindlessly.”