Officials at the Mexican facility refused to sign transfer papers until the family paid the bill, according to WFAA.
Family members said they hope to move Avila to a Dallas hospital.
Last year, the American Society of Plastic Surgeons estimated the average price for a nose job at $5,125 before factoring in anesthesia, outpatient operating room facilities, and other related expenses.
Experts at Texas Tech University Health Sciences Center in El Paso told KFOX the requirements to become a plastic surgeon in Mexico differ from those in the United States.
It takes around 16 years of extensive medical training to become an American plastic surgeon.