In dueling press conferences on Tuesday, President Barack Obama accused House Republicans of "extortion" and hostage taking. House Speaker John Boehner said that Obama wanted Republicans to unconditionally surrender.

And still, it looks like there's at least a small chance of a way out of the current mess in Washington.

Both Obama and Boehner seemingly left the door open to a short-term fix, in which the debt ceiling is lifted and the government is reopened — alongside a House-Senate deal that would establish a negotiating panel on fiscal issues.

Here's Obama, earlier on Tuesday (emphasis added):

"If there's a way to solve this, it has to include reopening the government and saying America is not going to default; it's going to pay our bills. They can attach some process to that that gives them some certainty that in fact the things they're concerned about will be topics of negotiation, if my word's not good enough.

"But I've told them I'm happy to talk about it. But if they want to specify all the items that they think need to be topic of conversation, happy to do it. They'll want to say, you know, part of that process is, we're going to go through, line by line, all the aspects of the president's health care plan that we don't like, and we want the president to answer for those things, I'm happy to sit down with them for as many hours as they want. I won't let them gut a law that is going to make sure tens of millions of people actually get health care, but I'm happy to talk about it."

Boehner didn't explicitly dismiss this idea in his press conference. And a "senior" House GOP source told CNN's Dana Bash that Republicans might be willing to jump on board with a short-term increase as long as Obama "agrees to use that time to negotiate." The extension could be for four to six weeks.

There is still a lot to be worked out. For one, it's unclear if any clean debt-limit increase — even with the promise of negotiations — can pass through the GOP-controlled House.

The White House also issued a veto threat to the GOP's plan to pass a bill that would establish a "negotiating team" among House and Senate Republicans and Democrats on the upcoming need to raise the debt ceiling. That's largely because the team would only recommend spending cuts from their negotiations, and wouldn't touch revenue. And the negotiations would be tied to the debt ceiling, something Obama won't accept.

Right now, a short-term fix seems like the best way out of the mess — even if it's far from a certainty.