The senator went on to warn that the GOP could see the loss or defection of a substantial number of conservatives over the perceived hostility of their immigration policy.

"Ultimately there is, in my opinion, hundreds of thousands of conservatives and potential conservatives all across this country that will never become conservatives because they somehow feel that the party where the conservative movement is housed doesn’t want them," Rubio said.

But Rubio defended his opposition to the DREAM Act, legislation popular among immigrant communities because it would reward young, successful and law-abiding immigrants with the opportunity to obtain legal status. The senator said "the support is not there" for a path to citizenship, but that there could be "a conversation" about legalization.

"I think most people would say that’s not amnesty, but it has to be structured in the right way. Then the other thing that I would say that’s wrong with the DREAM Act is it provides for chain migration, which is something people feel strongly about. It can’t be used as an anchor to let as many as 3 million people come into the country," Rubio said.

Rubio is Cuban-American, and has spoken before about the party's need to press the reset button on the immigration debate. Discussion of the topic will likely be among the main issues discussed at Wednesday's presidential debate in Arizona, where immigration activists have protested a state law that provided police with broader powers to question and detain those suspected of illegal immigration.