House Speaker John Boehner (R., Ohio) has yet to commit to any timetable for floor action, but has encouraged an effort by Rep. Mario Diaz-Balart (R., Fla.) to write a bill that attempts to strike a middle ground by giving legal status to most of the 11.5 million people in the U.S. illegally with the right to access existing pathways to citizenship. It rejects a “special pathway” to citizenship for this group, and envisions a string of hurdles that the government and individuals would have to cross before those provisions take effect.

By all accounts, Mr. Boehner wants to push forward on immigration legislation, knowing he will need a combination of Republican and Democratic votes to do so. Democrats are cautiously optimistic about the approach Republicans have laid out, but it is unclear whether Mr. Boehner can rally sufficient support inside his own conference.

There is uncertainty even within GOP leadership about the path ahead. Majority Whip Kevin McCarthy (R., Calif.) has said that midterm politics complicates any effort and that it makes more sense to wait, according to people familiar with the discussions. Majority Leader Eric Cantor (R., Va.) is seen as somewhere in between Messrs. McCarthy and Boehner.