House Speaker John Boehner said he doesn't want to run up against an August deadline for boosting the nation's debt limit - fearing that it could unnerve Wall Street - but he's refusing to back away from his calls for major spending cuts in return for a debt deal with the White House.

And Boehner, for the first time, said he wants to raise the limit within one month's time.

"I just think we're now in June, this really needs to be done over the next month if we're serious about no brinksmanship, no rattling investors," the Ohio Republican told a group of reporters in his Capitol office suite on Wednesday. "But I will reiterate something I've said many times before: The biggest risk we face as a country is doing nothing."

Boehner and the House Republican Conference met with President Barack Obama on Wednesday at the White House in a session that focused on the debt limit, deficits and job creation. Boehner said he told Obama that "the sooner we deal with" the debt limit, the "better off we are."

"The president agreed," Boehner added.

Yet Boehner's let's-get-a-deal-done stance masks a deeper belief within the House Republican Conference - that Obama will back down eventually and agree to its demands, forcing Capitol Hill Democrats to follow suit.

"Of course, it's dangerous," a House Republican close to Boehner said of the politics of a government default. "But it's dangerous for everybody, especially the president. At the end of the day, [Obama] will have to give in."

"Who has egg on their face if there is a sovereign debt crisis, House Republicans or the president?" asked another senior GOP lawmaker.

With a potential debt default by the U.S. government just two months off, and a continued standoff between the White House and GOP congressional leaders on how to move forward in boosting that limit, Republican lawmakers say publicly and privately that they believe Obama will be the one who has to cave.

Boehner and his top lieutenants - Majority Leader Eric Cantor (R-Va.) and Majority Whip Kevin McCarthy (R-Calif.) - are demanding that Obama put forth a comprehensive proposal to reduce the government's $1.5 trillion deficit, and they are refusing to move on a debt-limit boost until the president does so.

Despite what both sides called a "productive meeting" Wednesday, neither offered any new proposals to break the political stalemate.

And Budget Committee Chairman Paul Ryan (R-Wis.) challenged Obama to end political attacks by Democrats over Ryan's proposal to reform Medicare. Ryan complained that Obama has mischaracterized his proposal to replace the government-run system for senior health care with a privately administered program as a "voucher," according to several Republicans.

Obama countered by accusing the GOP of mischaracterizing his own proposals, suggesting Republicans had deliberately distorted his health care reforms, including a gradual $500 billion reduction in supplemental Medicare insurance purchased by seniors.

"We pressed him repeatedly to stop the demagoguery," Cantor told POLITICO after the White House session. GOP leaders also asked Obama, who was flanked by Treasury Secretary Timothy Geithner, to "stop saying that we don't have the best interests of the country at heart," Cantor added.

"I asked the president point blank, 'Are you going to put forth not a speech but a plan to deal with the debt crisis that is scorable by the Congressional Budget Office? What I got out of that was a resounding 'no,'" said Rep. Jeb Hensarling of Texas, who is chairman of the House Republican Conference.

"The demagoguery," Obama pointedly reminded Republicans, "comes from both sides."

Boehner acknowledged that the "bigger point" of the interaction between Ryan and Obama was delving into the "demagoguery" in Washington when lawmakers try to tackle huge policy issues.

"Trust me, it's happened from both sides of the aisle - I think Paul made that point perfectly clear, as did the president, I might add," Boehner said.

The view that Obama will be forced to cut a deal with Republicans is especially prevalent among freshman lawmakers, the driving force behind the GOP's focus on deficits and debt limits.

"There are simply not the votes to deliver this debt ceiling increase," said Rep. Adam Kinzinger (R-Ill.). "There are not. And there are people that have to be convinced to vote for it."

But the stakes are just as high for Boehner, who came under criticism from some conservatives for his agreement with Obama and Senate Majority Leader Harry Reid (D-Nev.) over the 2011 budget resolution.

While hailed by Boehner and his House allies as the best the Ohio Republican could get while avoiding a government shutdown, Boehner was criticized by some freshman Republicans for not doing enough. Boehner eventually had to call on House Democrats, led by Minority Whip Steny Hoyer (D-Md.), to pass the budget agreement.

Privately, some GOP lawmakers speculate that Boehner's position as the top House Republican could be undermined if he doesn't cut a deal that has wide support among GOP colleagues. "He could definitely be hurt if this goes wrong," admitted a Boehner supporter. "I don't see him being forced out. That won't happen. But he would lose the confidence of some."

Boehner has sought to avoid a repeat of what happened during the 2011 budget endgame as he takes on Obama and the Democrats on the debt limit. Although many GOP lawmakers don't believe Geithner - who travels to the Hill on Thursday to meet with Republican freshmen - when he says the U.S. government will be in default, triggering an economic meltdown, Boehner is not among them. Boehner acknowledges that the debt limit has to be increased at some point.

Boehner will not do so, however, unless Obama and Reid agree to "trillions of dollars" in spending cuts, a target that cannot be achieved without major changes to popular entitlement programs like Medicare and Medicaid. That stance carries political risks for Republicans, who lost a House special election last week in part because of the possibility of huge Medicare cuts under GOP plans to slash federal spending.

Boehner also wants to get this matter off his legislative agenda as soon as possible.

"I suggested to the president this morning, the sooner we deal with this, the better off we are," Boehner said Wednesday afternoon. "The president agreed. There's no reason to wait until whatever magical date somebody comes up with. We know we have to deal with this issue. We're going to deal with it as concerned Americans. So let's sit down and do it."

Boehner indicated that there has been "some progress" in bipartisan talks led by Vice President Joe Biden. Those discussions include Cantor, Sen. Jon Kyl (R-Ariz.) and top Democrats.

"They are making some marginal progress, but at the rate that that's going, we'll be right up against the wall," Boehner said.

But the speaker also opened the door to face-to-face negotiations directly with Obama as a way to move more quickly on the controversial issues.

"The president could engage himself. I'm willing; I'm ready; it's time to have a conversation," Boehner said, adding that there's no reason to bump against the August deadline. "I'm just ready to get on with it.

"It's time to play large ball, not small ball."