Why not attrition?
Most of those at the forefront of opposing mass immigration support "attrition". That means increasing immigration enforcement and decreasing benefits for illegal aliens, in order to encourage most illegal aliens to leave on their own.
Hundreds of thousands of illegal aliens left the U.S. during the economic downturn. Attrition would encourage even more to leave by making it more difficult for them to find jobs and access benefits.
Attrition is not at all the same as mass deportations: the goal of attrition would be for most illegal aliens to leave on their own rather than being forcibly removed.