"I think, no question, by the end of the year we could have a bill. No question," the speaker said.
In the Senate 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 GOP-led House to act.
Heated debate is anticipated on the border security elements of the bill. The bill sets up a system wherein immigrants may only begin taking steps toward citizenship once certain border security requirements are met. But opponents say those "triggers" aren't strong enough.
An amendment announced by Sen. John Cornyn, R-Texas., would require 100 percent monitoring of the entire U.S.-Mexico border and 90 percent of would-be crossers to be stopped or turned back before anyone can get a permanent resident green card. The Senate bill also sets those figures as goals, but doesn't make the path to citizenship directly contingent on them.
"My amendment demands results while the Gang of Eight bill is satisfied with just more promises, promises that historically have never been kept," Cornyn said Tuesday.
But in an interview over the weekend with Univision, Reid dismissed Cornyn's amendment as a "poison pill."
"If people have suggestions like they did in the Judiciary Committee to change the bill a little bit, I'll be happy to take a look at that," Reid said. "But we're not going to have big changes in this legislation."
It's not likely to be Cornyn's, but supporters of the bill were looking for a border security measure they could support. It could be an amendment pushed by Sen. Marco Rubio, R-Fla., an author of the bill who's talked about giving Congress a more direct role in developing a border security plan that the bill now leaves to the Homeland Security Department.
"One of the objections we've heard from the opponents of the bill is `we don't trust Homeland Security to come up with a plan that works.' Fine. Then let's put it in the bill," Rubio said Tuesday. "Let's put the specific plan in the bill, the number of fences, the amount of technology. Let's mandate it in the bill so that we're not leaving it to guesswork, so that when you vote for this bill you are voting for a specific security plan."
Other disputes will surround amendments being pushed by Sen. Orrin Hatch, R-Utah, to strengthen requirements for payment of back taxes in the bill and require previously illegal immigrants who get green cards under the bill to wait five years before beginning to access benefits under the nation's new health care law.
As debate moved forward in Congress, outside groups were ramping up their efforts, particularly supporters, who claim a larger coalition and deeper pockets. The Service Employees International Union announced plans to spend $1.1 million to run ads on cable stations nationwide to support the bill.