Rep. Mike Coffman, who represents a swing district in Colorado with about 20 percent Latino residents, and California Rep. David Valadao, whose district is 70 percent Latino, said they were both working with House Majority Leader Eric Cantor (R-Va.) on legislation for undocumented immigrants who came to the United States as children.

Coffman – a veteran of both Gulf Wars — said the so-called “Kids Act” would integrate parts of a bill he has introduced that would allow young undocumented immigrants to earn citizenship by serving in the military. The Kids Act is also expected to include education requirements.

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A Cantor spokesman said Wednesday there is no timetable to release the bill, which has been in the works for several months.

Speaking to reporters on Wednesday, Speaker John Boehner (R-Ohio) called immigration reform an “important subject that needs to be addressed,” and while he said he was “hopeful,” he was non-committal on a timeline for bills to be brought to the House floor.

Valadao added that he has had several conversations with the House’s top Republican, and Boehner has told him that he still expects a vote on immigration reform by the end of the year.

“I’m going to do my best to hold him to that,” Valadao said, later adding that “every conversation I’ve had with him tells me that he agrees we’ve got to be part of this conversation.”

Still, despite behind-the-scenes progress on writing new immigration measures, it’s unclear whether legalization bills drafted by Republicans would be able to pass the House, particularly if Democrats withhold their votes. A bloc of conservatives say they will oppose any immigration measure brought to the House floor.

In an interview Wednesday, Rep. Mario Diaz-Balart (R-Fla.), who has been working with other lawmakers on legislation that would deal with the broader undocumented population, said legalization bills will not be able to get support from a majority of House Republicans – required under the so-called Hastert Rule – unless border security and interior enforcement measures were strengthened.

One top House Democrat who has negotiated immigration measures with the Republicans in the past did not rule out providing Democratic votes for GOP legislation, depending on the policy.

“We are open to working with our Republican colleagues on anything so long as it is something real that could move the ball forward,” California Rep. Xavier Becerra, the fourth-ranking House Democrat, told reporters on Wednesday.

Still, other House Democrats were skeptical that the Republicans’ so-called piecemeal approach – which would overhaul sections of the U.S. immigration system in multiple bills, rather than one sweeping comprehensive measure – would work.

Instead, nearly all House Democrats have signed onto a bill that mirrors the Senate Gang of Eight bill, minus its border-security provisions. That legislation has no Republican co-sponsors.

“The idea that that same party who cannot pass anything … is now piecemeal going to do this is a fallacy,” said Rep. Joe Garcia (D-Fla.), one of the chief sponsors of that legislation.