President Obama’s outgoing Transportation Secretary Ray LaHood, a former GOP House member, says Republicans won’t win another White House race without a deal on major immigration legislation.

LaHood spoke to The Washington Post as he prepares to hand over the Cabinet job to Anthony FoxxAnthony Renard FoxxGeorgia Power says electricity at Atlanta airport will likely be restored by midnight Ex-Obama transportation chief on Atlanta airport power outage: 'Total and abject failure' To address America's crumbling infrastructure, follow Britain's lead MORE, the Charlotte, N.C. mayor that the Senate last week confirmed to replace him.

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“We will not win another presidential election as Republicans,” LaHood told the Post, “unless we do something to fix the broken immigration system.”

LaHood urged the House GOP to take a page from the book of Sen. Marco RubioMarco Antonio RubioRubio heckled by protestors outside immigration detention facility Bill to protect work licenses of student loan debtors is welcome development Political figures pay tribute to Charles Krauthammer MORE (R-Fla.), who helped craft the bipartisan plan the Senate approved last Thursday. If they don’t, LaHood warned, “this is going to be the death knell for Republicans winning national elections for decades.”

Some analysts see an immigration deal as crucial for Republicans’ national political viability going forward. Hispanic voters overwhelmingly favored President Obama in his race against GOP White House hopeful Mitt Romney last year.

Speaker John BoehnerJohn Andrew BoehnerFormer top Treasury official to head private equity group GOP strategist Steve Schmidt denounces party, will vote for Democrats Zeal, this time from the center MORE (R-Ohio) has said he will not shepherd an immigration bill through the House that does not have a majority of the chamber’s Republicans behind it, a threshold that complicates prospects for a House-Senate deal.

LaHood suggested that BoehnerJohn Andrew BoehnerFormer top Treasury official to head private equity group GOP strategist Steve Schmidt denounces party, will vote for Democrats Zeal, this time from the center MORE may need to buck his conference for the good of the party nationally.

“Look, this takes real courage,” he told the Post. “This takes the possibility of falling on [Boehner’s] sword for a greater good — for a greater good for the party, not for himself.”