The odds of the House passing an immigration reform bill are not "overwhelming," White House press secretary Jay Carney said Tuesday, after Republican leaders killed an amendment that would allow illegal immigrants to become legal permanent residents, and eventually citizens, by serving in the military.

"I'm not going to say the odds are overwhelming," Carney told reporters. "I mean, we're talking about House Republicans, and they have had a great deal of difficulty confronting this issue."

Last week, Republican leaders in the House announced they would block Rep. Jeff Denham's (R-Calif.) measure, which would amend the National Defense bill to offer a path to citizenship to those who served in honorably in the military. On Tuesday, House Speaker John BoehnerJohn Andrew BoehnerSpeculation mounts, but Ryan’s job seen as safe Boehner warns Trump: Don't pull out of Korea-US trade deal GOP Rep: Ryan wasting taxpayers dollars by blocking war authorization debate MORE (R-Ohio) said that the defense bill was not the appropriate venue for the measure.

"We have supported it in the past but trying to do this on the National Defense Authorization bill seems to us be an inappropriate place to do it," Boehner said.

The House Speaker said that while Republican leaders had discussed a stand-alone vision of the so-called ENLIST Act, no final decision had been made.

Still, the White House said it took solace in some Republican lawmakers advocating aspects of immigration reform — even if Carney maintained that Denham's legislation "would not in any way fix our broken immigration system or tackle the heart of the problem."

"We remain hopeful that they decide to act and act this year, because the benefits of comprehensive immigration reform are so clear," Carney said. "We know they won't act because the president wants them to act, but they might act because so many others that support comprehensive immigration reform are encouraging them to act."