Too many analysts keep putting the budget and debt fights into a partisan frame where Democrats are arguing with Republicans. In fact, it's a multi-level game where the President is fighting "Congress" -- notice that that is how he describes the other side, not "Republicans" -- and Democrats are fighting Republicans in two separate venues, House and Senate, which now and in the future fight each other.
I infer that the President is choosing his moves with an eye toward his reelection contest in 2012. He wants to lose as few voters from 2008 as possible, and motivate those who stayed home in 2010. He ran originally against conventional party politics and is trying to maintain that above-it-all stance. He needs deals on the debt ceiling, the deficit, and government programs for the sake of the country and to improve the economy over the next 18 months. And if problems arise, he wants everybody to recognize that he went the extra mile. I'm sure the White House is not indifferent to the needs of congressional Democrats, but those are far from their highest priority.
House Republicans think they have a mandate and should be able to dictate outcomes -- and they probably expect that the President will be blamed for whatever problems occur if they are intransigent. House Democrats want to win back control in 2012, as they could if they could recapture Obama-won districts, so they want to avoid any votes that trouble their base.
Senate Republicans are in very good shape to recapture the Senate, so they are happy to postpone tough votes, or measures that help the economy much. Senate Democrats are desperate to keep their majority and are individually quite divided over what makes the best sense politically and economically. Overall, congressional Republicans are fairly united and congressional Democrats split into numerous factions.
Given these many diverse pressures, and the absence of enough leaders who can resist partisan incentives, it's hard to predict the outcome right now. But I wanted to lay out the forces in play since the reportage has been so binary.