House Speaker John Boehner told "Fox News Sunday" that he's trying to put together a "framework" to deal with the deficit and debt ceiling by the end of the day Sunday, and that if Democrats won't sign on, the GOP will move forward unilaterally.

Boehner said he'd prefer to strike a bipartisan agreement with the other side of the aisle to both cut the deficit and raise the debt ceiling. But he said it's "too early to decide whether that's possible."

"I would prefer to have a bipartisan approach to solve this problem. If that's not possible, I and my Republican colleagues in the House are prepared to move on their own ... today," Boehner said.

The speaker walked away from White House-led talks Friday, accusing the president of pressing too hard for tax hikes. The next step, he said, is to craft a package modeled after the one that passed out of the House this past week. That plan, though, was widely opposed by Democrats who described its prescribed cuts as too drastic.

While Boehner said Sunday he's focused on "what's doable at the eleventh-hour," it's unclear how he might bridge the gap between the parties on anything modeled after Republicans' "cut, cap and balance" plan. The Senate already killed that plan once, and President Obama vowed to veto the original.

On a separate track, Boehner said his last offer remains "on the table."

Under that plan, Boehner acknowledged he had agreed to $800 billion in new revenue over the next decade -- combined with spending cuts and other changes worth far more than that.

Boehner insisted the $800 billion would come from a broadened tax base and not from any tax increase. "It was not raising taxes," he said, no doubt mindful of House conservatives' stiff opposition to anything that resembles a tax hike.

What shattered the talks, Boehner said, was a White House demand to weave $400 billion in additional revenue into the package.

At that point, Boehner said, "it was time for me to step back." He said he never took his last offer off the table, but added: "It may be pretty hard to put Humpty Dumpty back together again."

A White House official disputed Boehner's version of events. The official said the president asked for, but did not demand, $400 billion in additional revenue.

White House Chief of Staff Bill Daley, speaking on NBC's "Meet the Press," also said the administration merely asked Republicans whether they could accept more revenue in the package. "We never heard back," Daley said. "There was no ultimatum."

As both sides challenge each other's narratives over what went wrong, they continue to face an Aug. 2 deadline to strike a deal. The Obama administration says the country will face default after that date if the debt ceiling is not raised.

While Boehner said lawmakers are going to have to work on a "two-stage process" in the days ahead, the Obama administration has forcefully opposed any short-term deal that does not raise the debt ceiling through the next election.

Treasury Secretary Tim Geithner told "Fox News Sunday" that such a stopgap "makes no sense."