Washington publication Roll Call said Republicans Lamar Alexander of Tennessee, Kelly Ayotte of New Hampshire, Jeffrey Chiesa of New Jersey, Susan Collins of Maine, Bob Corker of Tennessee, Jeff Flake of Arizona, Lindsey Graham of South Carolina, Orrin G. Hatch of Utah, Dean Heller of Nevada, John Hoeven of North Dakota, Mark S. Kirk of Illinois, John McCain of Arizona, Lisa Murkowski of Alaska, Marco Rubio of Florida and Roger Wicker of Mississippi joined with Democrats in assuring bipartisan passage.
Majority Leader Harry Reid of Nevada said earlier if the Senate voted to end debate, the chamber likely would have a final vote on the bill by Friday.
"We're going to add immigration as another example of how to get things done," Reid said.
Sixty votes were needed to limit debate on the legislation, and the vote was held open for more than an hour to give several senators who experienced flight delays time to reach their desks, Roll Call said.
Sen. David Vitter of Louisiana and 13 other Republicans opposed to the bill sent a letter Monday to Reid calling on him to allow more time for debate and amendments.
"By point of reference, there were 46 roll call votes on floor amendments to the 2007 immigration bill, which was significantly stronger on enforcement," the GOP senators wrote.
"We believe that there should be, AT A MINIMUM, this same number of roll call votes on serious, contested floor amendments on the Gang of Eight's immigration bill. This can clearly be accomplished this week with a little leadership and coordination through one or more compact series of 10-minute votes with senators seated at their desks."
Sen. Rand Paul, R-Ky., said Sunday while the reform measure likely will easily pass the Senate it is doomed in the House.
"It will pass the Senate, but it's dead on arrival in the House," Paul told CNN's "State of the Union."
"The House is much closer to me, and I think they think border security has to come first before you get immigration reform," said Paul, a member of the Tea Party movement who describes himself as a "constitutional conservative" and a libertarian.
He said he remained opposed to the Senate bill and said the House agrees with him, despite a hard-fought border-security deal hammered out by Corker and Hoeven that Graham said would "almost militarize the border" by creating a "border surge" with double the border control force.
The Corker-Hoeven amendment would double the U.S. Border Patrol force to 40,000 agents and complete 700 miles of new border fencing, up from 350 miles lawmakers originally considered.
The additional border guards and fencing would cost roughly $25 billion, senators said.
The amendment also calls for a $3.2 billion high-tech border surveillance plan.
The plan includes drone aircraft, infrared ground sensors and long-range thermal imaging cameras. It would also require both an electronic employment verification program for all employers and a visa entry and exit system at all airports and seaports to limit immigrants who overstay their visas.
The amendment provides "more manpower, more fencing, more technology," Hoeven said Thursday.
Most senators say this strengthened border security is enough to sway a key number of Republicans earlier reluctant to support the broader immigration overhaul that includes a pathway to citizenship for 11 million immigrants in the country illegally.
Sen. Mike Lee of Utah, a charter member of the Senate Tea Party Caucus and one of the Senate's most conservative Republicans, told "Fox News Sunday" the amended bill was "likely to pass" with up to 70 votes.
But he said his vote would not be among the nearly 70.
"Look, we all want immigration reform. But this bill is not the right vehicle for getting there," he told Fox.
House Speaker John Boehner, R-Ohio, said at his weekly news conference Thursday an immigration overhaul bill would pass his chamber only if it ensured "our borders are secure" and gave "confidence that those who came here illegally are not given special treatment, confidence that hardworking taxpayers are being respected and confidence that a majority of both parties have had their say and support the final product."