WASHINGTON – Deeply divided, Republicans struggled without success Wednesday to find common ground over how to deal with the estimated 11 million immigrants who are in the U.S. illegally.

Republicans in the House of Representatives emerged from a 2½-hour closed-door meeting united in seeking tougher border security but with no solution for dealing with immigrants who already are here illegally.

“We didn’t decide anything,” said Rep. John Fleming, R-La. “We aired out our feelings.”

The party schism pits establishment figures such as former President George W. Bush and possible White House contender Sen. Marco Rubio, R-Fla., on one side and the party’s potent conservative base on the other.

The split was evident as the House Republicans huddle to plan their approach now that the Senate has passed an immigration overhaul that includes a path to citizenship for those who are in the U.S. illegally.

House Speaker John Boehner, R-Ohio, all but ruled out the kind of compromise plan that passed the Senate last month, saying he’d allow the House to vote only on a measure that a majority of Republicans supported.

The speaker “reassured people that, look, we’re not in a hurry here. We want to get something done — we think it’s very important to have a bill that passes — but we’re going to do it with a majority of the majority,” said Rep. Tom Cole, R-Okla.

After the meeting, Republican leaders issued a statement that asserted Americans “don’t trust a Democratic-controlled Washington, and they’re alarmed by the president’s ongoing insistence on enacting a single, massive, Obamacare-like bill rather than pursuing a step-by-step, common-sense approach to actually fix the problem.”

Just what the Republican alternative should be remains uncertain, and it was apparent Wednesday that to ease Republican tension, “We need a couple more conferences like this,” said Rep. Raul Labrador, R-Idaho.

Most lawmakers said that getting much done this month seemed unlikely, as they want to head home for the five-week August recess and talk to constituents.

Pressure to act is building from outside the Capitol, however, as Republican establishment figures and influential special-interest groups push hard for a legalization program.

They see political consequences: The Republican presidential nominee got 27 percent of the 2012 vote, and party officials fear they might lose even more support in the Latino community if they appear intolerant or insensitive.

Bush offered his views Wednesday at the George W. Bush Institute in Dallas. “We can uphold our tradition of assimilating immigrants and honoring our heritage of our nation built on the rule of law. But we have a problem,” he said. “The laws governing the immigration system aren’t working; the system is broken.”

An advocate of a broad overhaul of immigration laws, Bush enjoyed the support of 44 percent of Hispanics when he was re-elected in 2004. But he failed to get an overhaul through Congress, left office with his overall popularity hurt by unpopular wars and recession, and has little influence in the party.

The White House offered its own offensive Wednesday: The Obama administration released a report that shows the Senate bill would boost the U.S. economy and help create jobs.