The girl, whose name was not released to the public, had been accompanied by her mother. Both were transferred from the Mexican state of Chihuahua, which borders the US states of New Mexico and Texas.
"INM reports about the death of a 10-year-old girl from Guatemala who arrived in the migration center in Iztapalapa district in Mexico City on May 14," the statement said on Thursday, adding that the child died on Tuesday evening.
The child was diagnosed with a sore throat when she arrived at the Iztapalapa Migration Station, according to the INM, after which she was taken to a local children's hospital where she eventually died.
The Mexican authorities have opened an investigation into the case.
This girl was not the only migrant child who died at the southern US border on Tuesday. That day, a two-year-old Guatemalan boy died after being released from US Customs and Border Protection custody. He had trouble with breathing and ultimately was diagnosed with pneumonia. He became the fourth migrant child from Guatemala, where poor economic conditions are driving asylum seekers to the United States, to have died in US custody since December.
US President Donald Trump is highly concerned over flows of migrants coming to the United States through Mexico, insisting that a border wall should be in place in order to prevent criminals from entering the United States.