The reason, we are told, is a change that Republicans have previously rejected but Democrats and Latino groups see as indispensable: a procedure for illegal immigrants to become legal residents and, eventually, citizens. Critics call it "amnesty" and say it will spur illegal immigration as surely as emptying the jails would generate crime.
Warned one prominent opponent, Rep. Lamar Smith, R-Tex., "By granting amnesty, the Senate proposal actually compounds the problem by encouraging more illegal immigration." The Center for Immigration Studies (CIS) said it is a recipe for "effectively unlimited future illegal immigration."
This is not an entirely idle fantasy. Back in 1986, President Ronald Reagan signed a bill that eventually allowed three million unauthorized foreigners to become legal -- and in the following years, there was a surge of unapproved arrivals.
Today, the number of people living here without legal permission is estimated at 11 million. CIS Executive Director Mark Krikorian has said the amnesty option would resemble the experience of Bill Murray in the film "Groundhog Day" -- repeating an unwanted experience "over and over again."
The fears, however, are largely groundless. In the first place, the legalization program is not so generous as to lure hordes of Mexicans and other foreigners who have so far resisted the temptation to steal across the border. It would let undocumented migrants stay here on a probationary basis only after they pass a background check and pay a fine as well as back taxes.
So getting legal status won't be cheap or quick. Under the senators' proposal, those migrants yearning for actual U.S. citizenship would have to wait another decade or even two. It's amnesty only for the patient.
Of course, those who came in 1987 or 1993 have already been waiting a very long time for their shot at forgiveness. Is the prospect of another amnesty in 2040 really going to cause a stampede of unauthorized arrivals in 2014 or even 2024?
The alarmists also forget how much the world has changed since 1986. It's a mistake to blame the amnesty for the rise in illegal immigration that followed. It was due far more to a couple of other factors, according to Edward Alden, a researcher at the Council on Foreign Relations.
The first was a demographic bulge caused by Mexico's high birth rate: In 1960, the average Mexican woman had seven children. Today, the figure is down to nearly two. Mexico no longer produces enough people to export them in the volume it once did.
Another difference is the difficulty of getting here. "Border security is vastly more robust than before," Alden told me. Over the past two decades, he says, the number of Border Patrol agents has soared from 3,000 to 21,000, while 700 miles of fencing has been erected on our southern border.
Back then, walking in undetected was hard only if you had trouble walking. Today, the chances of arrest are high, and many aspiring entrants pay thousands of dollars to smugglers to improve their odds.
Fewer people take the risk these days. Despite having far more personnel, the Border Patrol apprehended fewer people in 2011 than in any year since 1972.
Not everyone wades the Rio Grande. Lots of the undocumented came here legally as students or tourists and overstayed their visas. But that tunnel has also gotten narrow.
The 9/11 attacks brought stricter checks and new controls on many foreigners coming here legally. In 2011, nearly two million visa applicants were rejected. That's nearly two million foreigners who couldn't become illegal immigrants.
Any comprehensive immigration measure won't ease those requirements. In fact, the senators insist on new steps to assure that those who come from abroad legally actually leave when their time is up. Nor will a reform bill turn the Border Patrol into the Welcome Wagon.
But the critics are right: If illegal immigrants gain a path to legal status and citizenship, the change is likely to evoke "Groundhog Day." In the movie, every day begins the same way. What changes is what happens next. In the end, remember, they live happily ever after.