It also offered up four other general principles it wants to see in an immigration bill. These include reforms to address backlogs in the immigration process, reforms that are "consistent with civil, constitutional and basic human rights," a commitment to support civic and economic engagement by immigrants, and "smart, effective and humane enforcement measures" at the border.

"Both Jewish and Latino immigrants have struggled to overcome discrimination and find a balance between integrating into the American culture and preserving their rich heritage," caucus members said Monday. "Our communities have built strong coalitions across a number of issues, based on this common historical experience; nowhere is that more evident than in our mutual work on immigration issues and reforming our nation's broken immigration system."

Rep. Ileana Ros-Lehtinen (R-Fla.), one of the leaders of the caucus, said an immigration bill supporting these principles would be "consistent with our American principles."

"We are a nation of laws, and we are also a nation based on opportunity and fairness for all," she said. "I am hopeful that my colleagues in the House can keep these principles in mind as we work together to honor the rule of law and strengthen our families and our communities."

Another caucus leader, Rep. Eliot Engel (D-N.Y.), called on the House to move legislation that creates some way for illegal residents to win legal status.

"We cannot ignore the 11 million people – and their families, which include American citizens –living in our country," he said. "Fixing this situation will strengthen our national security, aid our economy, and end the exploitation of millions of people."

But despite these calls, House GOP leaders continue to insist that the House will do its own immigration bills, and have not said they would pursue any bill creating a pathway toward legal status. Over the weekend, House Speaker John BoehnerJohn Andrew BoehnerWe need more congressional oversight on matters of war A warning to Ryan’s successor: The Speakership is no cakewalk With Ryan out, let’s blow up the process for selecting the next Speaker MORE (R-Ohio) declined to say whether the House would consider the idea, and immigration bills that have moved so far deal mostly with enforcement and visas for high-skilled workers.