The way forward in the House remains unclear, because the chamber is dominated by conservatives many of whom view a path to citizenship as amnesty. But Boehner said he hopes for committee action by the end of June.
"I believe that it's important for the House to work its will on this issue," Boehner said on ABC's "Good Morning America." ''And I would expect that a House bill will be to the right of where the Senate is."
"I think, no question, by the end of the year we could have a bill. No question," the speaker said.
As debate opened Tuesday in the Senate, Minority Leader Mitch McConnell, R-Ky., said the bill written by a so-called Gang of Eight — four Republicans and four Democrats — would need stronger provisions on border security and other issues to earn his support.
"The Gang of Eight has done its work. Now it's time for the Gang of 100 to do its work — for the entire Senate to have its say on this issue, and see if we can do something to improve the status quo," McConnell said. "At the risk of stating the obvious, this bill has serious flaws."
Nonetheless McConnell said he intended to vote to allow the debate to go forward. The real fights will come in the days and weeks ahead as Senate Majority Leader Harry Reid, D-Nev., aims to push the bill to final Senate passage before July 4.
The bill's Senate supporters were working to determine which amendments they could accept to lock down more "yes" votes from the GOP side without losing Democratic backing. They are aiming for a resounding show of support from the Democratic-led Senate that could pressure the House to act.