WASHINGTON (AP) — President Barack Obama is using his presidential pulpit to press the Republican-controlled House to embrace a path to citizenship for all 11 million people living illegally in the U.S., while a top Republican says those brought to America as children should be given the highest priority for legalization.

With prospects shaky for passing an immigration overhaul in the House, the White House insisted Monday that to garner Obama's signature, any bill must satisfy the president's principles — the path to citizenship chief among them. But Obama is leaving the particulars of how Congress gets there up to lawmakers, wary of strong-arming the process and handling Republicans an excuse to vote no.

"I cannot even begin to count the number of possibilities that could emerge through the House process. So I'm not going to," said White House spokesman Jay Carney. "What I'm saying is that the end result has to meet the president's principles if he's going to sign it into law."

He said Obama would voice those principles and the benefits of fixing immigration Tuesday in interviews with Spanish-language TV stations.

If the White House had its way, the House would simply and swiftly take up a sweeping Senate bill that passed with bipartisan support. But House Speaker John Boehner has already rejected that notion, preferring to tackle the nation's immigration laws in "bite-sized chunks."

Boehner's approach reflects the intense skepticism of the GOP rank-and-file, who say they don't trust Obama will fulfill border-security requirements in the Senate-passed bill. Although Republicans generally acknowledge they must broaden their appeal to Hispanic voters whose influence in elections is rapidly growing, many say they fear primary challenges from the right if they support a new path to citizenship.

One exception that could gain traction among Republicans would be to offer a citizenship path only to those brought here as children. Allowing only those individuals to obtain citizenship could shield Republicans from attacks by conservatives that they're giving a free pass to those who voluntarily broke the law.

"I think that group of people — some call DREAMers — is a group that deserves perhaps the highest priority attention," Rep. Bob Goodlatte, who chairs the House Judiciary Committee, said at an immigration-related conference in California. "They know no other country."

Goodlatte and House Majority Leader Eric Cantor, both Virginia Republicans, are working on a bill to address the status of those immigrants, although the timing is uncertain. And Goodlatte cautioned that any such measure should hinge on completion of enforcement measures to prevent parents from smuggling their children into the U.S. in the future.

Obama has kept his distance from the back-and-forth in Congress, knowing that the political imperative for Republicans will likely be more persuasive than any pressure he applies — and that aligning himself too closely to any legislation could actually make it harder to attract GOP support.

Still, with a major component of his legacy on the line, Obama is ratcheting up his efforts to make clear to Republicans that without meaningful action on immigration, Republicans will be hard-pressed in future elections to peel Hispanic support away from Democrats. More than 70 percent of Hispanics backed Obama in his re-election last year.

From the grounds of the White House, Obama will take to the airwaves Tuesday of Spanish-language TV stations in Dallas, Denver, Los Angeles and New York. The White House said Obama will argue that immigration reform is in line with the nation's values and in the country's economic interests.