Republicans warned Democratic Senate Majority leader Harry Reid was driving the bill too fast, and said failure to consider their objections could kill the measure, which could shape the destiny of millions of immigrants and is strongly supported by President George W. Bush.
But the top Senate Republican Mitch McConnell said many members of his party, some of whom are facing a severe backlash over their support for the bill, wanted to table amendments.
"Before they would allow us to wrap up this bill, these amendments would need to be considered," McConnell said.
Reid warned Republicans should not seek to use delaying tactics, and try to blame him for any consequent failure of the bill.
"I would hope that my Republican colleagues are not going to use this as an excuse ... that really is not fair, it is wrong," Reid said.
Reid was adamant that he would hold a vote on ending debate on the bill on Thursday, leaving it unclear if he could get the necessary 60 votes in the closely divided chamber to proceed.
Should his bid to end debate fail, he said he would shelve the bill, likely dealing a death blow to the legislation, as any attempt to bring it up later this year or next year would see it caught in the maelstrom of the 2008 congressional and presidential elections.
The fragile coalition of lawmakers behind the deal hopes to cling together through the storm of amendments, many of which appear designed to kill the measure.
The bill, agreed last month with the White House, would also establish a merit-based points system for future immigrants and institute a low-wage temporary worker program.
It includes a border security crackdown, punishments for employers who hire illegal immigrants and an attempt to wipe out a backlog of visa applications from those who have gone through legal immigration channels.
Bush last week accused opponents of the bill of trying to stir up fear among Americans.
"If you want to scare the American people, what you say is, 'The bill's an amnesty bill.' "It's not an amnesty bill."
Several amendments have been tabled to attempt to reorient the bill more towards family-based immigration, from its emphasis on merit- based applications.
Earlier Tuesday, senators overwhelmingly endorsed an amendment which would require US firms to seek to hire Americans under the guest worker program, before seeking to take on foreigners.
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