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.
Authorities apprehended Couch and his mother, Tonya Couch, last month in Puerto Vallarta, Mexico.
Tonya Couch is charged with hindering the apprehension of a felon and could face up to 10 years in prison if convicted. Ethan Couch is still being held in Mexico as he fights deportation.
Tonya Couch, madre del joven prófugo detenido en México, declaró que no puede pagar la fianza de un millón de dólares impuesta por un juez de Texas por ayudar a su hijo a escapar
Ethan Couch remains in custody in Mexico after winning a delay on his deportation back to Texas.
Tonya Couch, who was deported from Mexico last week, arrived on a flight from Los Angeles and was taken to Tarrant County Jail in handcuffs and leg irons.
El martes, un juez de Los Ángeles ordenó que Tonya Couch, madre del joven fugitivo detenido en México, regresara a Texas para que las autoridades locales determinen su situación
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.