The Guatemalan girl, who arrived in the US last week as part of a group of migrants, died of dehydration and shock, the Washington Post reports.
AP news agency quotes border officials as saying she had not had food or water for several days.
Thousands of migrants have travelled from Central America to the US border.
The migrants say they are fleeing persecution, poverty and violence in their home countries of Guatemala, Honduras and El Salvador.
Many of them say their goal is to settle in the US despite warnings by US officials that anyone found entering the country illegally will face arrest, prosecution and deportation.
What do the US authorities say?
The US Border Patrol confirmed that an accompanied girl aged seven had died after experiencing fever and seizures while in its custody for eight hours on Thursday of last week.
The girl, whose nationality was not given, was flown to hospital in El Paso where she suffered cardiac arrest and died within 24 hours.
A Department of Homeland Security spokesperson sent condolences to the girl’s relatives, saying US Border Patrol border “always takes care of individuals in their custody and does everything in their power to keep them safe”.
“Unfortunately, despite our best efforts and the best efforts of the medical team treating the child, we were unable to stop this tragedy from occurring,” the spokesperson added.
“Once again, we are begging parents to not put themselves or their children at risk attempting to enter illegally.”
On Thursday, the Washington Post reported that the girl had died after being detained along with her father.
Tension has been running high on the US-Mexican border since the arrival of almost 7,500 migrants in recent weeks.
Last month, US border agents used tear gas on a crowd of migrants, including children, trying to cross the border.
The agents said that personnel had been assaulted and hit by stones.
However, critics accused the Trump administration of a draconian response, while Mexico demanded an investigation into the incident.
The migrants have travelled in large groups, dubbed “caravans”, for more than 4,000km (2,500 miles) from Central America.
Among them are many families with young children.
Donald Trump has vowed to keep each migrant on the Mexican side of the border until courts have decided their cases, meaning some face a long wait.
They have been spending time in temporary shelters in the Mexican border city of Tijuana and in Mexicali, 180km to the east.