NEW YORK (

TheStreet

) -- President Barack Obama visited El Paso, Texas on Tuesday and addressed the growing debate about immigration reform in his first trip to the U.S.-Mexico border since assuming office.

The Democratic president is looking to put pressure on Republicans to undertake comprehensive immigration legislation as an estimated 11 million illegal immigrants live in the United States today.

Obama insisted that the country must "keep up the fight to pass genuine comprehensive immigration reform," citing America's legacy as a nation of immigrants.

"We define ourselves as a nation of immigrants -- a nation that welcomes those willing to embrace America's precepts," Obama said.

In his speech, Obama challenged the call from Republicans to focus on border security, saying that their demands have been more than met by his administration but that "they'll never be satisfied."

"We have gone above and beyond what was requested by the very Republicans who said they supported broader reform as long as we got serious about enforcement," Obama said. "But even though we've answered these concerns, I gotta say I suspect there are still going to be some who are trying to move the goal posts on us one more time."

"Maybe they'll need a moat," Obama said mockingly, evoking laughter from the crowd. "Maybe they'll want alligators in the moat."

He said reform is needed to find a solution for the millions of undocumented workers in the critical to the country's economic future.

White House spokesman Jay Carney told reporters traveling on Air Force One to El Paso that Obama recognizes the legislative hurdles involved in taking on the issue and is now "trying to build awareness and support for the need for comprehensive immigration reform to fix this broken system."

Obama said that, after reform, illegal immigrants would first be held responsible for paying fines for breaking the law and then expected to pay.

He also said the new legislation would hold businesses accountable if they choose to exploit undocumented workers.

Obama is also looking to gain support from the growing Democratic Hispanic population, as its continued support would bolster his 2012 re-election bid.

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Written by Theresa McCabe in Boston

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