“Well, it's certainly a precondition,” said the Texas lawmaker. “It's got to pass with strong momentum in the Senate to have a chance in the House.

“Nothing really original I think is going to originate in the House of Representatives.  So that's really a precondition,” he added. “If it does that, I actually think it has a good choice.  I still believe that we can pass it in 2013.”

Efforts to pass immigration reform in the Senate were bolstered this week after an amendment to toughen border security measures, a sticking point for conservatives, was negotiated by Sens. Bob CorkerBob CorkerThe Hill's 12:30 Report Iran nuclear deal still under threat — US must keep its end of the bargain Senate heads to new healthcare vote with no clear plan MORE (R-Tenn.) and John HoevenJohn HoevenMcCain absence adds to GOP agenda’s uncertainty McCain diagnosis looms over GOP healthcare talks This week: ObamaCare repeal faces latest setback in Senate MORE (R-N.D.).

Reform advocates believe the measure could sway more GOP support and create momentum to push an immigration bill in the House, where it faces an uphill climb.

But Speaker John BoehnerJohn BoehnerSudan sanctions spur intense lobbying OPINION | GOP's 7-year ObamaCare blood oath ends in failure A simple fix to encourage bipartisanship in the House MORE (R-Ohio) has said he will not move any bill that lacks the support of a majority of the GOP conference.

Castro warned that adhering to the so-called Hastert Rule, named after the former speaker, could kill immigration in the House.

“That means that 25 percent of the body can control 100 percent of the agenda and the legislation,” said Castro. “It will not pass if he uses the Hastert Rule.”

Rep. Mike Kelly (R-Pa.), though, called on his fellow lawmakers though to be cautious before endorsing a broad overhaul of the nation's immigration laws.

"We talk about a sovereign nation and its ability to protect its borders.  That's number one.  If we didn't learn anything from 1986 up until now, it's that we've got to look at this carefully and go through it," he said, also on ABC.

"I don't understand the rush.  We saw what happened in the Patient Protection and Affordable Care Act.  Any time you rush anything through that big -- this was up to 1,100 pages -- I doubt that anybody's really read it and been able to really get through every -- every piece of it."