After budget fight, Boehner says he and Obama "understand each other better"

House Speaker John Boehner said in an interview on Fox News this morning that after the protracted battle over this year's budget, he and President Obama "understand each other better."

"Clearly, after spending the last five or six weeks in pretty close contact with the president, I think we've understood more about each other," he said. "We've developed a process that may allow the debate to go forward. But understand, ideologically there are giant differences between the president and myself."

Boehner added, "It's our obligation to sit down and try to find a way to work together."

While Boehner and Mr. Obama may have found a way to work together to forge a budget compromise, the House speaker is continuing to take shots at the Obama administration. In a USA Today op-ed published today, Boehner said the House will give its "full attention" this week to the 2012 proposed budget put forward by Republican House Budget Committee Chairman Paul Ryan (even as work on this year's budget is finalized). Dubbed the "Path to Prosperity," Ryan's plan promises to cut about $6 trillion in government spending over the next 10 years.

"This week, we'll advance our fight from saving billions of dollars to saving trillions of dollars," Boehner wrote. Ryan's plan, he says, "leads where the administration has failed and takes on autopilot spending that's driving our debt crisis."

"The budget by Chairman Ryan has set the bar," he continues. "If the president is willing to follow our lead and offer serious proposals that address the drivers of our debt and the barriers that are holding back our economy, we'll welcome it, and we're open to hearing them."

Mr. Obama has already proposed his own 2012 budget, but Boehner called it "irresponsible." The president now seems to be heeding calls for more aggressive entitlement reform. On Wednesday, he plans to lay out his own idea for Medicare and Medicaid reform.

Meanwhile, Boehner challenged the president over the looming vote on raising the debt ceiling over the weekend. According to projections by the Treasury Department, the U.S. government is expected to hit its $14.3 trillion debt ceiling by the end of May. If Congress does not approve an increase to the limit, the federal government could default on its bonds for the first time in history, and Social Security and Medicare checks would likely see delays as a result of the government's inability to make payments to agencies.

"There will not be an increase in the debt limit without something really, really big attached to it," Boehner said at a weekend fundraiser.

"The president says I want you to send me a clean bill," he said. "Well guess what, Mr. President, not a chance you're going to get a clean bill."

On CBS News' "Face the Nation" Sunday, Democratic Sen. Chuck Schumer said bargaining over the debt limit "is playing with fire - because if the markets believe we are not going to pay our debt, it could be a formula for recession or worse."

Watch as Democratic strategist Jamal Simmons and Republican strategist Doug Heye discuss the budget battle with CBS News political producer Rob Hendin on CBSNews.com's "Washington Unplugged." The roundtable also discussed the president and Republican leadership's next fight: whether or not to raise the debt ceiling.