May 20, 2014 1:39 PM EDT

House Republicans are prepared to spike a proposal that would allow children brought into the country illegally to gain legal status through military service, in the latest sign that immigration reform is a no-go ahead of the midterm elections.

Rep. Jeff Denham (R-Calif.) introduced the ENLIST Act as an amendment to the annual defense bill on Monday , along with with Rep. Tammy Duckworth (D-Ill.), an Iraq War veteran. But he earned a swift rebuke from his party leaders. House Speaker John Boehner and Majority Leader Eric Cantor, who have previously voiced support for the measure, will not allow the amendment to be considered with others on Tuesday.

Boehner said Tuesday that while “we have supported it in the past,” it would be “inappropriate” to attach it to the National Defense Authorization Act. When asked if the ENLIST Act would receive stand-alone vote, Boehner said there have been discussions but no decision.

Denham said the measure is appropriate to include in the larger defense bill.

“It’s a change to military code, not immigration law,” Denham said in an open statement Monday. “The ENLIST Act would allow otherwise qualified undocumented immigrants brought here by their parents through no fault of their own to earn legal status through military service.”

Democrats, who broadly support a path to citizenship for undocumented immigrants, have blasted GOP leaders for not allowing a vote on the amendment.

“It is clear that Rep. Eric Cantor is the one standing in the way of immigration reform,” Pili Tobar, a Democratic National Committee spokesperson, said in a statement Tuesday. “The ENLIST Act would provide a path to citizenship for DREAMers who are willing serve, fight and put their lives on the line for this country, but that’s still not enough for Republican leadership. Republicans keep paying lip service to the issue but they’re all talk and no action. This legislation should be a no brainer.”

Rep. Luis Gutiérrez (D-Ill.) said Tuesday on the House floor that if Republicans don’t help pass immigration reform in 2014, George W. Bush will be “the last Republican president in American history.”

Republican leaders across the country fear what’s been called a demographic death spiral with Hispanic voters, as a growing bloc of Americans increasingly backs Democratic candidates.

But individual Republican lawmakers face pressure from voters in their conservative base, making it almost certain the House won’t move on the issue before this year’s midterm elections. The Senate passed a comprehensive immigration reform bill last year that has since withered in the House, and Boehner reiterated his conference’s resistance to comprehensive reform on Tuesday.

“Now listen, we’ve talked about this literally every week for the last 18 months, and I’ve made it clear over the last several months that until the President gives us some reason, some confidence that we can trust him to implement an immigration reform bill, we have nothing to talk about,” Boehner said. “The ball is in the President’s court.”

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