The Mexican drug boss will file multiple legal challenges against Friday's ruling to extradite him to the U.S.
Ethan Couch, 18, arrived at Dallas-Fort Worth International Airport late Thursday morning and could be seen walking through the airport escorted by law enforcement.
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.
Authorities apprehended Couch and his mother, Tonya Couch, last month in Puerto Vallarta, Mexico.
A federal judge admits an appeal against two extradition processes against him.
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.
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.