Cruz's spokeswoman said the language of the amendment hasn't been drafted, but she made clear that it "will specify that the federal Motor Voter law does not preempt state laws that require ID for voter registration."
"The intent is to be sure states can pass laws ensuring only citizens can vote in order to limit voter fraud," Cruz spokeswoman Catherine Frazier said in an email. "To make sure people are who they say they are."
The amendment to immigration legislation has the potential to divide pro-reform senators, as the issue of voter ID laws tends to break down largely along party lines. Cruz has supported elements of reform to overhaul and expand legal immigration but strongly opposes a path to citizenship for people in the country illegally.
The 1993 National Voter Registration Act requires states to accept a federal form that lets people register to vote by attesting under oath that they are citizens. The Court's decision Monday said it preempts state laws like the Arizona statute in question.
Cruz promoted his upcoming amendment on Twitter:
I'll file amendment to immigration bill that permits states to require ID before registering voters & close this hole in fed statutory law.— Senator Ted Cruz (@SenTedCruz) June 17, 2013