And in fact, there is reason to believe that many of them could be harder to convince today than they were before the election. The reason: While the party lost nine seats on Election Day, the rest of its caucus was actually reelected by bigger margins than in 2010.
According to a Fix review, more than one-quarter of Republicans elected to the House in 2010 (61 out of 242) took less than 55 percent of the vote. In the 2012 election, that number was down to one-sixth (39 out of 234).
That's not all that surprising. Much of it has to do with partisan gerrymandering and also the fact that many Republicans were running as incumbents for the first time in 2012.
But regardless, it has got to make many of them feel safer than they were before -- and perhaps more concerned about their primaries than the general election.
Suburban Philadelphia Reps. Jim Gerlach, Mike Fitzpatrick and Pat Meehan, for example, all took at least 57 percent of the vote despite coming from swing districts. Obama was in their area last week pitching his vision for avoiding the fiscal cliff.
Elsewhere, moderate Rep. Dave Reichert (R-Wash.) had his best showing ever with 60 percent of the vote in a revamped district, and top freshman targets like Reps. Sean Duffy (R-Wis.), Reid Ribble (R-Wis.) and Scott Tipton (R-Colo.) all won by double digits.
The question is whether these members feel greater pressure in the primary (which might push them away from voting for a deal that includes tax increases) or in the general election (which might push them toward such a deal).
If you're someone like Fitzpatrick, Meehan, Gerlach, Duffy or Ribble, you have to be feeling better about your future prospects in the general election, given that you just won a sizable victory in a good Democratic year.
Winning by double digits is pretty solid ground for an incumbent, and taking 55 percent or more of the vote is generally the cutoff between who is considered vulnerable and not-so-vulnerable.
The fact that five-sixths of House Republicans took more than 55 percent of the vote in 2012 suggests they might be more concerned about losing their jobs in the primary season than in the general election.
There is one major caveat.
Because we are in the lame duck session, Republicans members who lost or are retiring will play a major role in determining whether a deal can be reached. And there are 41 of them.
Since they won't necessarily be running again, many of them won't be worried about their future political prospects -- either in the primaries or the general election -- and a couple have already signaled a willingness to allow tax cuts for the wealthy to expire: Reps. Robert Dold (R-Ill.) and Mary Bono Mack (R-Calif.).
But not having to face reelection could also work the other way and embolden them to reject a deal. Reps. Joe Walsh (R-Ill.), Ann Marie Buerkle (R-N.Y.) and Allen West (R-Fla.), for example, won't have to worry about voting their conservative consciences in their Democratic-leaning districts anymore.
The current breakdown in the House is 241 Republicans and 192 Democrats, meaning Democrats need to get at least 25 Republicans to support a deal.
How the members mentioned above vote will say a lot about whether a deal gets done.