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Mr. Coffman said he still may try to offer his policy as an amendment.

“I have not ruled out putting an amendment on the NDAA that would allow these young people the chance to serve this country in uniform and earn a path to citizenship through their military service,” said Mr. Coffman, a Marine Corps combat veteran and member of the committee.

House leaders haven’t taken a stand, but one aide said they are comfortable with how Mr. McKeon is handling the debate.

National Guard Association of the United States has backed Mr. Coffman’s bill. A spokesman says the association believes it should play a role as Congress debates defense issues this year, particularly because House Republicans are unlikely to move legalization legislation on its own.

“We understand why some believe it could be a distraction, but we continue to believe the bill offers a pathway for immigration that needs to be part of the broader debate,” said John Goheen, communications director for the association.

The American Legion, however, said it not want to mix immigration and defense policy and opposes granting citizenship rights to illegal immigrants in the first place.

“The legion’s long-standing policy remains that we are opposed to any policy, any legislative action that amounts to amnesty, and I think that would fall under that definition,” Mr. Stovall said.

The issue is apparently too hot for some other groups.

Neither Concerned Veterans for America nor Amvets responded to repeated requests for comment. Veterans of Foreign Wars declined to comment.

Mr. Denham’s bill would have the military accept illegal immigrants who were brought to the U.S. before age 15 and who were in the country before 2012.

Mr. Coffman’s bill builds off Mr. Obama’s 2012 non-deportation policy for “dreamers,” the name young illegal immigrants have given themselves. Mr. Coffman’s legislation says anyone who receives a work permit under that non-deportation policy can join the military and get in line for citizenship.

That could be even tougher to pass than Mr. Denham’s amendment because it would represent an official congressional approval of those non-deportation policies, said Rosemary Jenks, government relations director for NumbersUSA, which advocates a crackdown on illegal immigration.

Whether an amendment could win approval in committee is in doubt. Mr. Coffman’s bill doesn’t have many Republican co-sponsors, and none of them other than Mr. Coffman is on the committee. Mr. Denham’s bill has about a half-dozen Republicans on the committee as co-sponsors, but it’s unclear whether they would approve of mixing the immigration debate with the military policy debate.

That likely means the issue ends up back at the feet of House Republican leaders, who eventually have to decide whether to allow a floor debate on the issue.

“I don’t know that there would be enough Republicans to pass it in committee. On the floor, that’s a whole different thing,” Ms. Jenks said. “If Denham insists on offering his amendment on the floor and [Majority Leader Eric] Cantor tells him he can, first of all I think there’s a huge bloody fight and it very possibly passes. But all of this depends on, [House Judiciary Committee Chairman Bob] Goodlatte signing off on the language and Eric Cantor driving it.”

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