The girl and her father had been detained by immigration authorities on December 6 in New Mexico as part of a group of 163 people who approached US agents to turn themselves in, the Post reported on Thursday.
Early on December 7, the girl started having seizures, and emergency responders measured her body temperature at 40.9C, the Post said. She was taken to a hospital, where she died, according to the Post.
The names of the girl and her father were not released. The agency, which typically provides food and water to migrants in its custody, is investigating the incident to ensure whether appropriate policies were followed, the Post said.
The head of the Department of Homeland Security, which oversees Customs and Border Protection, will appear in front of the House of Representatives Judiciary Committee next week, Jerry Nadler, the top Democrat on the panel, said on Twitter.
"We will be demanding immediate answers to this tragedy," Mr Nadler said.