The terrible political damage Republicans sustained in the government shutdown fight is apparently making some House Republicans even less willing to pass immigration reform, because mistrust of Obama is running higher than usual. To me, that’s a bit like saying: We just shot ourselves in the foot, so let’s get back at the president by shooting ourselves in the other foot. But for some Republicans, this is where we’re headed.

However, immigration reform still isn’t quite dead yet.

In an interview, GOP Rep. Mario Diaz-Balart of Florida told me that a “number” of House Republicans are in negotiations to develop a piecemeal solution to the problem of the 11 million undocumented immigrants — with the goal of commanding a majority of Republicans. This is good to hear. It means not all Republicans are using the shutdown loss as a way to bail on immigration reform.

“There are a number of us who are working on a proposal to deal with the folks who are here in a way that allows those who have not committed crimes to get right with the law,” Diaz-Balart says, adding that the goal is to figure out “what to do with the millions of undocumented who are here in a way that completely conforms with the rule of law.”

Diaz-Balart declined to specify who these Republicans were or what policy fixes are being discussed. But he provided a clue to the thinking. Republicans insist they can’t support anything that doesn’t secure the border before permitting legalization, and will only accept a piecemeal approach.

So one model for a proposal to deal with the 11 million in a piecemeal way could be an idea drawn from the now-defunct House “gang of seven” talks.  House Republicans could pass border security measures, and then pass a measure that puts the 11 million on probation — status they would lose if E-Verify is not operational after five years. Legalization can only proceed after it is operational.

“It is a non-negotiable objective to make sure we have border and interior security as part of anything we do,” Diaz-Balart said. “It has to be enforceable. That aspect that was agreed to [by the gang of seven] is an example of something that’s workable.”

“We have to get the majority of Republicans in support, but on something this difficult and controversial, we’re going to need Democratic votes as well,” he continued. That’s a key point, because it suggests those in these talks recognize that, because the far right will not support anything, the solution must be crafted to be acceptable to Dems, too.

Conservatives are also gearing up to block House Republicans from going to conference on immigration — again in part because Obama and Dems can’t be trusted to negotiate. I asked Diaz-Balart to respond to that argument and whether he thought Republicans should be willing to go to conference.

“To me that’s a bit of a technicality. The question is, do we want to move forward on legislation to fix the borders? We’re going to have to take some political arrows but that’s what we’re here to do,” he told me. “I’m hearing a lot that you’ve got a president you can’t trust or negotiate with. That’s pretty much a consensus among Republicans. Here is what is also consensus: We have porous borders and 11 million here unlawfully. The broken system is not something Republicans should accept. If we don’t solve this issue, it will be back year after year. The number of undocumented will continue to grow.”

If enough House Republicans agree with the above sentiment, immigration reform is not dead yet.

Indeed, in addition to the need to repair relations with Latinos, reform’s chances may turn on whether enough House Republicans — and their leadership — decide they want to be seen as a party that is capable of solving the country’s problems. Today’s Post/ABC poll found that only 20 percent of Americans think the GOP is “interested in doing what’s best for the country.” Now, maybe such considerations just don’t matter, because the House GOP majority is supposedly invulnerable. But if Republicans do want to broaden the party’s appeal and show they are capable of constructive governing, immigration reform is one of only a handful of ways to do this.

Greg Sargent writes The Plum Line blog, a reported opinion blog with a liberal slant -- what you might call “opinionated reporting” from the left.