"That will be the great crucible," Sen. Charles Schumer of New York, the lead Democratic negotiator, said of the legislation's test before the Judiciary Committee.
What has happened in the interim is the 2012 election.
Many Republicans believe that their stance against a pathway to legalization for illegal immigrants became so politically toxic that it alienated Latino voters, who constitute a rapidly growing segment of the electorate.
"This time around, there's been a sea change on the Republican side," Graham said.
Despite that, some senior Republicans still worry that the legislation will crumble under its own weight by trying to fit in so many pieces of the issue.
"The difficulty is trying to pass a comprehensive bill. Because the more things you put together, the more opponents you develop," Sen. Lamar Alexander, R-Tenn., said.