Under the House plan, many immigrants brought to the country as children, agricultural workers and highly skilled immigrants would likely be granted permanent residency. Then presumably they could apply for citizenship within a handful of years, as under the current system. How many years is unclear.

Many conservative Republicans also are likely to be dissatisfied with the House proposal.

Republican Rep. Steve King of Iowa still hasn’t gotten over the so-called 1986 amnesty bill signed by Republican President Ronald Reagan. It legalized 3 million undocumented immigrants and failed to prevent a wave of millions more from rushing into the country.

“The promise then was law enforcement in exchange for amnesty,” King said Wednesday during a Capital Hill discussion on immigration among conservative Republicans. “We have the same dialogue going on now that we had in 2006 and 2007. And I’m sitting here a bit astonished that we’re seriously considering that our president is going to help us enforce immigration law.”

At the same meeting, Labrador said any new pathway to citizenship was “off the table.”

He told McClatchy afterward that Democrats needed to come to the right “a little bit.”

“I think if they continue to insist on it, then it could potentially blow up the entire immigration reform package,” he said.

The bipartisan House group has been meeting for months behind the scenes to craft principles before working on the details of more specific legislation.

The core members started talking in 2009, then essentially went into hibernation a year later as the vitriol over immigration reached its peak. They began meeting again after last fall’s elections, picking up new members such as Labrador

For Diaz-Balart, the work has been a family affair as he followed in the footsteps of his brother, former Rep. Lincoln Diaz-Balart, R-Fla., in pushing the issue in the face of strong Republican Party opposition.

“I’m absolutely sure that we have the ability and the willingness to get this done,” he said this week.

The House leadership has issued its support for getting an agreement. House Speaker John Boehner, R-Ohio, has applauded the work of House members and said Congress must deal with immigration this year.

House Majority Leader Eric Cantor, R-Va., said last week that he hoped Congress could follow labor and business – which has agreed on a set of principals for immigrant workers – in putting politics aside to find common ground on the immigration front.

“Let’s hope we can follow that lead in the months ahead,” he said in a statement.