Washington, AP—US senators are preparing to cast the first votes in the full Senate on a landmark bill that offers the best chance in decades to remake the American immigration system and offer eventual citizenship to millions.

President Barack Obama, who’s made overhauling immigration laws a top second-term priority, was to speak at a midmorning event with advocates at the White House to praise the Senate’s efforts and renew his calls for reform. Immigration reform represents a chance for Obama to gain a big legislative victory amid several distracting scandals.

Heated debate is anticipated on the bill’s border security elements. The bill sets up a system wherein immigrants may only begin taking steps toward citizenship once certain border security requirements are met. But opponents say those “triggers” aren’t strong enough, and one of the bill’s authors, Sen. Marco Rubio, a rising Republican star from Florida, has argued that the bill’s border security elements must be strengthened if it’s to make it through Congress.

After overwhelmingly losing the Latino vote in last November’s elections, several Republicans have shown a willingness to work with Democrats to fix what both parties criticize as a broken immigration system. But other Republicans reject any plan that would lead to citizenship for immigrants living in the US illegally.

Some Republican senators said they were seeking to strengthen enforcement provisions so that they could be comfortable voting for the bill. Other Republican measures were already being dismissed by Democrats as attempts to kill the bill by striking at the fragile compromises at its core.

Ahead of Tuesday afternoon’s procedural votes to officially allow debate to move forward, senators were readying amendments on contentious issues including border security, back taxes and health care coverage.

The bill’s supporters were working to determine which measures they could accept to lock down more “yes” votes from the Republican side without losing Democratic backing. They are aiming for a resounding show of support from the Democratic-led Senate that could pressure the Republican-led House of Representatives to act.

The two votes scheduled for Tuesday afternoon were on procedural measures to officially allow debate to move forward on the far-reaching bill. Both votes were expected to succeed by comfortable margins, because even some senators with deep misgivings about the immigration bill said the issue deserved a Senate debate.

The real fights will come in the following days and weeks as Senate Majority Leader Harry Reid aims to push the bill to final Senate passage before July 4.

Even if that happens, the outlook in the House remains unsettled, but House Speaker John Boehner has indicated he’d like to see a bill through his chamber before August.

The Senate bill would stiffen border security and require all employers to check their workers’ legal status, as well as initiate new or expanded visa programs for high-skilled and lower-skilled workers and the agriculture sector. At its core is its most contentious element, a 13-year path to citizenship for some 11 million immigrants now here illegally.

“Given the impact the broken system has on our economy and our families, we cannot afford delay,” Senate Judiciary Chairman Patrick Leahy said Monday on the Senate floor. “This is a measure the Senate should come together to consider and pass.”

An amendment announced by Republican Sen. John Cornyn would require 100 percent monitoring of the entire US-Mexico border and 90 percent of would-be crossers to be stopped or turned back before anyone can get a permanent resident green card. The Senate bill, authored by a bipartisan group of eight senators, also sets those figures as goals, but doesn’t make the path to citizenship directly contingent on them.