Republicans said they would try to toughen border security provisions and impose tougher terms on people seeking legal status before the final vote on the legislation. Senate Minority Leader Mitch McConnell (R-Ky.) said the 60 votes likely required would not be assured.
The comprehensive immigration bill seeks to create a path to citizenship for 11 million people living in the United States illegally.
Republican Florida Senator Marco Rubio said about half of the Senate's 46 Republicans are prepared to vote to create the pathway to citizenship. That aim is backed by nearly all 55 lawmakers aligned with the Democratic majority in the Senate.
At its core, the bill sets out a 13-year journey to citizenship for the millions of immigrants who arrived in the United States illegally through the end of 2011 or who overstayed their visas. That journey would include paying fines and back taxes and other measures. The bill also requires a tighter border to prevent future illegal immigration.
Other key provisions would create a new program for low-skilled workers to enter the country and expand the number of visas for high-skilled who are particularly in demand in technology firms. The bill also jettisons a decades-old system that favors family ties over education, job skills and other factors in prioritizing prospective legal immigrants.
Referring to the 11 million currently in the country unlawfully, President Barack Obama said: "Yes, they broke the rules; they didn't wait their turn. They shouldn't be let off easy. They shouldn't be allowed to game the system. But at the same time, the vast majority of these individuals aren't looking for any trouble. They're just looking to provide for their families, contribute to their communities."
Numerous Republicans hope to use the issue to repair their party's image among Hispanic voters, a growing portion of the electorate in key states, and a group that polls show gave Obama 71 percent of its votes last year. But the GOP is divided, with tea party-backed lawmakers and other conservatives resisting anything that smacks of amnesty or otherwise seems to permit legalization without assuring the long border with Mexico in particular is virtually closed to future unlawful immigration.
U.S. House of Representatives Speaker John Boehner (R-Ohio) said, "I think, no question, by the end of the year we could have a bill. No question." In the Senate, McConnell sounded a similar note.
The Associated Press contributed to this report.