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.
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.
Ethan Couch remains in custody in Mexico after winning a delay on his deportation back to Texas.
The mother and son were arrested in the Mexican resort city of Puerto Vallarta late last month.
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.
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.
Tonya Couch, 48, appeared for an extradition hearing in downtown Los Angeles, where she was flown last week after being deported from Mexico.