Senate Majority Leader Harry Reid has set up the final procedural hurdle to clear the immigration reform bill for votes Thursday and Friday.
The Senate will vote around noon Thursday to invoke cloture or cut off debate on the entire bill.
The White House-backed bill would pour billions into border security and offer a path to citizenship to some 11 million immigrants now in the United States illegally.
The bill already has sidestepped several procedural obstacles with votes to spare, demonstrating it commands well over the 60 votes needed to pass the Senate.
Once cloture is invoked, the Senate rules allow opponents of cloture up to 30 hours of debate, meaning the final vote might not come before Friday afternoon.
"A permanent, common-sense solution to our dysfunctional system is really in sight," said Reid, D-Nev. "It is my hope that our colleagues in the House will follow the Senate's lead and work to pass bipartisan reform and do it now."
Opponents weren't convinced.
"It continues to promote false promises that the border would be truly secure," said Sen. Charles Grassley, R-Iowa.
Prospects were anything but clear in the GOP-controlled House, where many conservatives oppose citizenship or even legalization for people in this country illegally. House Speaker John Boehner, R-Ohio, said he has no plans to hold a vote on the legislation coming from the Senate.
"We are not going to take up the Senate bill," Boehner told Republicans in a morning meeting, according to Rep. John Fleming, R-La.
In the Senate, after the addition of $38 billion in provisions strengthening border security, doubling the size of the border patrol, and completing hundreds of miles of fencing, the legislation looked likely to command support from more than a dozen Republicans on final passage.
That's more than enough to ensure the 60-vote margin needed for passage, as all 52 Democrats and the two independents who usually vote with them look likely to stick together.
Supporters rounded up 67 or more "yes" votes on each of three tallies Wednesday. One vote overcame a Republican budget objection, the second was to approve changes to the bill including billions in new border security spending, and the third allowed moving forward with the amended version of the bill.
Supporters dodged a potential obstacle when the Supreme Court on Wednesday struck down a provision of a federal law denying federal benefits to married gay couples. That would allow gay married Americans to sponsor their spouses for U.S. residency in the way straight married Americans can.
Gay rights supporters had been pushing for the immigration bill to include such a provision, but Republicans had warned it would sink the immigration bill.
The Supreme Court's ruling made it unnecessary, which was the outcome supporters were hoping for.
Fox News' Chad Pergram and The Associated Press contributed to this report.