A judge will now consider the deportation of Ethan Couch, 18.
He and his mother were arrested in the resort town of Puerto Vallarta after a phone call for pizza tipped off police.
Couch left Texas for Mexico after he allegedly broke probation. His lawyer had argued a privileged upbringing left him with no sense of responsibility.
An arrest warrant was issued earlier this month for Couch after he failed to report to his probation officer.
His disappearance came shortly after he had apparently been filmed at a party where people were consuming alcohol, police said.
He could go to prison for up to 10 years if found to have violated his probation by drinking.
His mother Tonya Crouch faces charges of hindering an apprehension.
She and her son planned their disappearance and even held a going-away party, said police.
It is not yet clear if she has also sought to delay her extradition.
They were being held at immigration offices in Guadalajara and had been due to fly to Houston on Wednesday evening.
Couch was on juvenile probation after killing four people in a 2013 drink-driving crash in 2013.
He pleaded guilty and a judge sentenced him in juvenile court to 10 years' probation and a stint in rehab.
A psychologist at his trial successfully argued that his privileged upbringing - an unrecognised condition known as "affluenza" - meant his parents had not taught him a sense of responsibility.
A fugitive who made headlines for using an "affluenza" defence after a fatal drink-driving crash has delayed his extradition from Mexico to the US.