This power rests with the INS, which must back the policy which means that the Administration must back the policy.
That is not going to be easy, as the press, liberal lawyers and judges and of course the DNC will wage a propaganda war that will make their war against Joe McCarthy seem like small potatoes.
The only way to clean up the illegal mess is incrementally.
The first step is to to deny a drivers license to anyone who is not a citizen, a power that the States do have.
It has already been implemented in Florida and elsewhere post 9-11, and is being resisted in Democrat conrolled states like California.
People without a license can then be held as possible illegals and turned over to INS and deported in small batches without a lot of fanfare.