A new visa program for low-skilled workers would be developed and the Department of Homeland Security would be charged with coming up with a $4.5 billion plan to gain effective control of the porous Southern border with Mexico.

A final vote on the measure, after the amendment process, is expected to come before July 4. Even if it passes, it faces a cloudy fate in the Republican-controlled House.

Regardless, the bill is getting a major push from President Obama, who said Monday that the nation’s immigration system “hasn’t kept pace with changing times and hasn’t matched up with our most cherished values.” He urged lawmakers to vote for the bill, calling it “a common-sense, bipartisan bill that is the best chance we’ve had in years to fix our broken immigration system.”

“Right now, our immigration system keeps families apart for years at a time,” he said. “Even for folks who, technically, under the legal immigration system, should be eligible to become citizens but it is so long and so cumbersome, so byzantine, that families end up being separated for years.  Because of a backlog in visas, people who come here legally — who are ready to give it their all to earn their place in America — end up waiting for years to join their loved ones here in the United States. It’s not right. But that’s the broken system we have today.”

Rubio, who hails from a Cuban family and is thought to have White House aspirations, continues to provide mixed signals about his intentions even though he was a prominent member of the “Gang of Eight,” the bipartisan group that pieced the compromise package together.

Recently, Rubio has joined fellow conservatives in voicing concern that the bill doesn’t sufficiently address border security. He has indicated that he won’t vote for it in its current form despite earlier pledges.

“Border security is not an anti-immigrant measure,” he said. “I refuse to accept the idea that the most powerful nation on Earth, the nation that put a man on the moon, is incapable of securing its own border.”

Sen. Ted Cruz (R-Texas), one of the bill’s primary opponents, said he nonetheless supports the concept of immigration reform. Americans are convinced, he said, that the current system is not working and that the nation needs to secure the borders and improve and streamline legal immigration.

“On those basic principles, there is widespread bipartisan agreement, and if this body were to focus on those areas of bipartisan agreement, that’s how we would get an immigration bill passed into law,” he said.

But Democrats and the White House are standing in the way, he said, because they are insisting on a provision that provides a path to citizenship for the undocumented workers already here “and threw everything else overboard.”

“I very much hope we work together in a bipartisan manner to fix this problem in a way that secures the border, in a way that respects rule of law and in a way that improves legal immigration so we remain a nation that welcomes and celebrates legal immigrants,” Cruz said.