Under this idea, people here illegally would be allowed to live and work in the U.S., and could then apply for green cards, which are hard to obtain but serve as a prelude to citizenship. The bill passed by the Senate this year would make most of the 11 million people here illegally eligible for green cards—or legal permanent residency—after a set period of time, which automatically gives someone the chance to apply for citizenship.

“How many of the 11 million would take this deal rather than nothing?” asked Tamar Jacoby, president of ImmigrationWorks USA, a business group that backs an immigration overhaul. At a forum last week, she challenged Republicans to offer this deal and Democrats to accept it.

“How will Democrats who reject this deal explain themselves to Latino voters?” she asked.

The challenge of passing immigration legislation was underscored in recent days when two Republican House members abandoned an effort to work with Democrats to write a broad, bipartisan bill. The departure of Reps. John Carter and Sam Johnson, both Texas Republicans, means three of the four GOP lawmakers involved have now left the bipartisan group, signaling its likely end. The group had been working in secret for four years to try to craft a bill.