Sen. Rand Paul, R-Ky., said his proposal would require Congress to vote annually for five years on whether the border is secure. If lawmakers decide it is not, "then the processing of undocumented workers stops until" it is, he said. The decision would be made based on numerous factors, including progress toward completion of a double-layered fence along the U.S.-Mexico border and toward a goal of 95 percent capture of illegal entrants. A system to track the border comings and goings of foreigners is also required.
Only a day earlier, the CBO had cheered supporters of the bill with an estimate that it would help the economy and reduce deficits in each of the next two decades.
Now it was the skeptics' turn to crow.
"Illegality will not be stopped, but it will only be reduced by 25 percent," said Sen. Jeff Sessions, R-Ala., referring to the prediction by the non-partisan CBO.
While the public debate was taking place, lawmakers involved in the private talks expressed optimism.
"We're on the verge of doing something dramatic on the border," Graham told reporters. "What we're trying to do is put in place measures that to any reasonable person would be an overwhelming effort to secure our border. This is a key moment in the effort to pass the bill."
Across the Capitol, House Republican leaders sought to present a friendlier face to Hispanics - a group that gave Obama more than 70 percent support in last year's presidential election.
Boehner met with the Democratic-dominated House Hispanic Caucus, while rank and file members of his party reviewed areas of agreement with faith-based Latino leaders.
"It's a conversation Republicans want to have," Rep. Cathy McMorris Rodgers, R-Wash., said later at a news conference outside the Capitol.
At the same time, though, anti-immigration protesters moved across the Capitol plaza into range of television cameras, raising signs that said, "Do Not Reward Criminals" and "No Amnesty for Illegal Aliens."
Separately, the House Judiciary Committee worked on legislation creating a program allowing farm workers to come to the United States to take temporary jobs in the United States.
The measure is one of several that the panel is considering in the final weeks of June as part of a piece-by-piece approach to immigration rather than the all-in-one bill that Senate is considering.
In addition to border security measures and a pathway to citizenship for millions of immigrants in the country illegally, the Senate bill provides more visas for highly-skilled workers prized by the technology industry, a guest worker farm program and a new program for lower-skilled workers to come to the United States.