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.
El venado se encontraba aproximadamente a 200 metros de las playas de Mismaloya; tras su rescate fue llevado a un zoológico para que médicos veterinarios valoraran su estado de salud
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.
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.
Ethan Couch and his mother, Tonya Couch stayed first at a resort known as Los Tules, near the beach.
During the sentencing phase of his trial, a defense expert argued that his wealthy parents coddled him into a sense of irresponsibility - a condition the expert termed "affluenza."
Ethan Couch, 18, had been a fugitive after breaking his probation sentence for killing four people while driving drunk. He was arrested in Puerto Vallarta.
Las mejores recomendaciones para disfrutar de la playa y la buena comida