There is one thing the Republicans want, which is for the White House to start offering them what they want without having to demand it themselves. The White House has done this sort of negotiating before. In fact, it was their template in the stimulus bill negotiations; the health care bill negotiations; the financial services bill negotiations; the Bush tax cut extension negotiations; and the debt ceiling negotiations. In each case, the Democrats would offer something designed to attract Republicans, they would happily take it and then demand even more. In each case, Republicans got far more than they would have gotten ordinarily and far more than they deserved. Since last September, the president no longer negotiatiates like that. If Republicans want something they're going to have to demand it themselves. President Obama is refusing to negotiate with himself.
The key question is this: When will Republicans get specific about cutting spending?
If they don't then it should be clear to all that the Republicans aren't serious about cutting the deficit.