The parents of Jakelin Caal said the girl had been given food and water and appeared to be in good health as she travelled through Mexico with her father, 29-year-old Nery Gilberto Caal Cuz.
In a statement released by lawyers, the family also said Jakelin had not spent days in the desert before she was taken into custody, as officials had claimed.
Tekandi Paniagua, the Guatemalan consul in Del Rio, Texas, said Mr Caal told him the group they were travelling with was dropped off in Mexico about a 90-minute walk from the US border.
Border Patrol officials did not immediately respond to the family’s comments. A cause of death has not yet been released.
On Friday US Customs and Border Protection (CBP) said Jakelin had a temperature of 41 degrees and had not eaten or drunk anything for several days.
The family’s statement was released Saturday during a news conference in El Paso, Texas, at an immigrant shelter where Jakelin’s father is now staying.
Jakelin and her father were seeking asylum in the US and were among a large group of migrants arrested 6 December near a border crossing in New Mexico.
Hours later they were placed on a bus to the nearest Border Patrol station, but Jakelin began vomiting and eventually stopped breathing. She later died at a Texas hospital.
Border Patrol officials said agents did everything they could to save the girl. They added that an initial screening showed no evidence of health problems, and that her father had signed a form indicating she was in good health.
But the family took issue with that form, which was in English, a language her father doesn’t speak or read. He communicated with border agents in Spanish but he primarily speaks the Mayan Q’eqchi’ language.
“It is unacceptable for any government agency to have persons in custody sign documents in a language that they clearly do not understand,” the statement said.
A private prayer service was held in Texas on Friday so her father could see Jakelin’s body before it is taken to Guatemala, said Ruben Garcia, director of the Annunciation House shelter where her father is staying.
“All of us were moved by the depth of his faith and his trust that God’s hand is in all of this,” Mr Garcia said.
Family members in Guatemala said Mr Caal decided to migrate with his daughter to earn money he could send back home. Jakelin’s mother and three siblings remained in San Antonio Secortez, a village of about 420 people.
Additional reporting by Associated Press