At its contentious core, though, is a 13-year path to citizenship for the 11 million immigrants living in this country illegally.
Without such a provision, senators say the legislation could not pass the Senate. With it, its prospects are difficult in the House.
Sen. Chuck Schumer, D-N.Y., predicted that the House might end up having to pass the Senate bill after failing to find any other avenue forward and feeling pressure from the public to act.
But that approach is strongly opposed by many conservatives. Boehner also dismissed the idea of relying on Democratic votes instead of a majority of his Republicans to pass an immigration bill.
At the same time Boehner said he hopes the bill will be bipartisan, and he encouraged a House group of four Democrats and three Republicans trying to forge a compromise to continue their efforts.
He offered no details on how a House bill could be both bipartisan and supported by more than half of his own rank and file, given that most of the single-issue immigration bills that have moved through the House Judiciary Committee recently did so on party-line votes over the protests of Democrats. None envisions legal status for immigrants now here illegally.
Speaking alongside Diaz-Balart Friday, Rep. Zoe Lofgren, D-Calif., a top Democrat on the Judiciary Committee, dismissed the bills passed by her committee so far as "a series of small-bore, partisan bills that are in some cases bizarre."
Boehner declined to say whether there were circumstances under which he could support a pathway to citizenship, but he made clear that securing the border was a priority.
"People have to have confidence that the border is secure before anything else is really going to work. Otherwise, we repeat the mistakes of 1986," he said, referring to the last time Congress overhauled the immigration system.
One option could be to bring up one or more of four narrowly focused immigration bills approved by the Judiciary Committee this week and last, hoping to pass it and use it as a vehicle for House members to enter into negotiations with senators on a merged bill in the fall or winter.