Thompson said on CBS' "Face the Nation" he also wants to require employers to use "modern technology" to determine who they are hiring and to "clamp down" on sanctuary cities by cutting off federal funding.

"... If you did all those things, there would be attrition over a period of time," Thompson said.

"Nobody's advocating that these people all be rounded up. Nobody's advocating that they even be arrested and incarcerated, although they're in violation of the law," he said. "They're simply saying that over a period of time that if you enforce the law and secure the border, there will be attrition that will make things start moving in the other direction."

Thompson said he believes if illegal immigrants know they can't go back and forth across the border and aren't protected from the law in certain cities, "they'll go back on their own accord."

"Obviously, not all of them and not overnight," Thompson said. "But it will move in the right direction."