Ethan Couch fled to Mexico in December with his mother after apparently violating the probation deal reached in juvenile court that kept him out of prison for killing four people in 2013.
A Mexican federal judicial official said that Couch's attorney in Mexico filed a document on Sunday seeking to lift the injunction that has kept the 18-year-old in that country.
Ethan Couch's legal team presented a document to a court in western Jalisco state seeking to end the effort.
Even if a judge agrees that Ethan Couch was forced to flee to Mexico, he could still be locked up if the judge determines he violated probation by attending a party where people were drinking.
Ethan Couch remains in custody in Mexico after winning a delay on his deportation back to Texas.
Ethan Couch was driving drunk and speeding near in June 2013 when he crashed into a disabled SUV, killing four people and injuring several others, including passengers in his pickup truck.
Lawyer Fernando Benitez said he will meet with client Ethan Couch to determine whether Couch stretches out the process, or agrees to drop the appeal.
Cruz, the son of a Cuban immigrant, wants to dramatically increase deportations, add hundreds of miles to the wall on the Mexican border and reverse every immigration order signed by President Barack Obama.
In 1954, the operation ("Operation Wetback") was aimed at apprehending and deporting agricultural workers who had crossed the border illegally looking for work.
Most of the foreign inmates are Mexicans, according ICE.