Senate Majority Whip Dick Durbin invited Minority Whip Jon Kyl on Sunday to privately discuss ways to move forward on comprehensive immigration reform.

“Here’s the offer I’ll make to Jon Kyl and [Sen. John McCain (R-Ariz.)]: I will sit down with you, I will work with you to have border security beyond what the president’s done — and I think we’ve done an enormous, made an enormous investment there,” Durbin (D-Ill.) said while appearing with Kyl (R-Ariz.) on “Fox News Sunday.”

Kyl had just said on the program that the nation needs “to turn off the faucets of illegal immigration, and then we can turn to these other issues” of addressing the estimated 11 million illegal immigrants already living in the United States.

Durbin added, “I will go even further to make sure that our border is safe and to stop, as much as humanly possible, illegal immigration, if you will join with us in comprehensive immigration reform so that we can identify those living in this country, we can give the children under the DREAM Act an opportunity to have a good life in this country and we can finally fix this broken system.”

Senate Republicans blocked the DREAM Act on a procedural vote in December, and Durbin and other Senate Democratic leaders reintroduced the measure last week. The bill would create a path to citizenship for immigrants who were brought to the United States before age 16 and have been in the country for more than five years if they go to college or serve in the military.

“In good faith, let’s put both on the table, and let’s agree to get it done before the end of these two years,” Durbin concluded.

Kyl, who has announced he will not run for re-election in 2012, accepted the invitation.

“Dick Durbin and I sat down before to talk about things, and I’m very happy to do it on this very important issue, but I will say we’ve got to secure the border in order to achieve these other results as well,” he said. “But thank you, Dick, I’d be happy to sit down, and you know we can visit together about these things.”