President Barack Obama on Tuesday correctly hailed the “good news” of a rising probability that Congress will finally approve comprehensive immigration reform this year.

A day earlier, a bipartisan group of eight senators, including S.C. Republican Lindsey Graham, presented their own general framework for such a landmark bill.

Many Republicans, though not Sen. Graham, have only recently come around on this issue. Cynics — or merely realists? — trace that conversion to the continuing growth in both Hispanic voters and the Democrats’ winning-percentage margin from that group.

Both ballot-box trends were resoundingly re-confirmed last November: The president’s re-election victory, according to a consensus of exit polls, included a roughly 69-29 percent edge among Hispanics.

But while politicians inevitably have political motives, common sense can also occasionally move them toward common ground.

That is reflected by the Senate “Gang of Eight,” which includes Republicans Graham, John McCain and Jeff Flake of Arizona, and Marco Rubio of Florida, and Democrats Michael Bennet of Colorado, Richard Durbin of Illinois, Robert Menendez of New Jersey, and Charles Schumer of New York.

And conservatives, moderates and liberals alike seem increasingly ready to come together on how to better handle the 11 million illegal immigrants already in the U.S. — and how to gain better control of our southern border.

As President Obama said during Tuesday’s speech in Las Vegas: “At this moment, it looks like there’s a genuine desire to get this done soon. And that’s very encouraging.”

Still, Sen. Graham, in a statement released Monday, warned that if Congress doesn’t pass immigration reform this year, “it will be many years before anyone is willing to try and solve this problem.”

Thus, he urged, “We should start this new attempt hopeful and with full understanding how difficult the task is.”

Certainly the man whom Rush Limbaugh dubbed “Lindsey Grahamnesty” knows transforming obsolete immigration policies is a tough job. In both 2006 and 2007, Sen. Graham joined Sen. McCain in championing President George W. Bush’s attempts to achieve it. Those efforts ultimately failed, largely due to a public outcry that drove strong opposition from the right.

Yet Sen. Graham also recalls that Mr. Obama, then a senator from Illinois, undermined the 2007 initiative from the left. Sen. Graham told Fox News’ “On the Record” Monday night:

“If you want to talk about the president bringing us together and being a bold leader, in 2007 when we tried to do immigration reform, he folded like a cheap suit when labor ... got mad at the bill to allow temporary workers. Obama, Senator Obama, put a sunset on the temporary worker provision, which destroyed the bill for business.”

But that was then. This is now.

And as Sen. Graham said Monday, “I hope the third time is the charm.”

Meanwhile, before condemning the reform plan as “amnesty,” despite its requirements that illegal immigrants pay fines, learn English, get jobs and get in the back of the line for citizenship applications, ponder this lingering reality:

Rounding up and deporting 11 million people — many of whom fill crucial labor needs — is not a practical option.

However, the status quo is unacceptable, too. It would perpetuate a bewildering limbo for not just illegal immigrants but U.S. citizens, businesses, law enforcement, and municipal and state governments.

In the glaring absence of a coherent federal policy, several states, including our own, have tried to cope with the mess by writing immigration laws of their own.

But our national government remains the proper entity to solve this national problem.

So now it’s up to Congress, presumably with President Obama’s backing of a balanced, bipartisan result, to deliver a worthy compromise for long-overdue immigration reform.