Washington (CNN) -- A compromise amendment intended to ease congressional passage of sweeping immigration legislation easily won Senate approval Wednesday.
The 69-29 vote set up a procedural motion immediately afterward to limit debate on the roughly 1,200-page bill, which would provide an eventual path to citizenship for millions of immigrants living illegally in the country.
While some further amendments still could be considered, the measure appeared headed to a final Senate vote by the end of the week after the procedural motion passed in a 67-31 vote.
Backers of the bill want the Democratic-led Senate to pass it with a solid majority to demonstrate growing bipartisan momentum as the measure heads to the GOP-controlled House of Representatives.
The compromise amendment by Republicans John Hoeven of North Dakota and Bob Corker of Tennessee would increase border security, a demand by conservative opponents of the immigration bill.
"I hope our colleagues in the House of Representatives will follow the Senate's lead, and work to pass bipartisan reform that both Democrats and Republicans can support," Senate Majority Leader Harry Reid, D-Nevada, said before Wednesday's votes.
Conservative GOP Sen. Chuck Grassley of Iowa, an opponent of the Senate bill, conceded it s likely to pass and called on House Republicans to make things right.
"I expect the House of Representatives to fix this miserable failure," Grassley said.
Republican leaders say the House will consider the issue in several smaller proposals instead of a comprehensive single measure, a process that would allow more debate and votes on specific provisions.
Earlier Wednesday, House Speaker John Boehner reiterated to GOP members that he opposes having the House take up any Senate immigration legislation, according to Rep. John Fleming.
Fleming, a Louisiana Republican, said Boehner told the weekly policy meeting that the House would work on its own immigration bills.
If enacted, the bill would create a path to citizenship for roughly 11 million undocumented immigrants while enhancing security along the border with Mexico.
Among other things, the compromise border security amendment would add 20,000 border agents, complete 700 miles of fence along the boundary with Mexico, and deploy $3.2 billion in technology upgrades similar to equipment used by U.S. forces in Iraq and Afghanistan.
The proposal also includes stronger worker eligibility verification standards and border entry-exit controls.
Most undocumented immigrants would be eligible for permanent residency only after those five conditions have been met and verified by the Department of Homeland Security.
Hoeven and Corker introduced the compromise amendment last week, saying it incorporated proposals from other senators to try to fix a broken immigration system.
Grassley and other opponents argued the legislation amounts to an amnesty for immigrants who broke the law by living illegally in America. They argued the promises of increased border security before giving undocumented immigrants permanent residency would end up proving empty.
"It does nothing to change the legalization-first philosophy and does little more than offer false promises that the American people can no longer tolerate," Grassley said.
If eventually passed into law and signed by President Barack Obama, the bill would be the first major immigration reform since 1986, during the second Reagan administration.
Grassley conceded he voted for the 1986 measure, which also called for increased border security, and called his support then a mistake he regrets.
The 2013 version was drafted by a Senate "Gang of Eight" composed of four Democrats and four Republicans motivated by political and policy needs.
Both parties concede the nation's immigration system is broken, and some Republicans believe that GOP refusal to work with Democrats on the matter would mean a repeat of the 2012 presidential election in which Obama won a strong majority of the Hispanic vote.
However, other Republicans fear that providing a path to legal status for millions of undocumented immigrants would bolster support for Democrats from the new voters.
Florida Sen. Marco Rubio, the most prominent Republican in the "Gang of Eight," told CNN's Dana Bash last week that he understands "why conservatives are upset."
"They've seen all these promises in the past that haven't been delivered," Rubio said. But the status quo on immigration "is hurting America," he added. "And if nothing passes, then this disaster that we have now, that's what's going to stay in place."
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CNN's Alan Silverleib, Ted Barrett and Deirdre Walsh contributed to this report.