Police said they were unsure when the boy, who had been given the name Adrian, would be reunited with his mother, who is in Mexico, the El Paso (Texas) Times reported.
"We have located the mother and working on an exchange," the department said on its Facebook page. "We will keep you all posted!"
Police Chief Jaime Reyes said the decision is up to the New Mexico Children, Youth & Families Department.
A Border Patrol agent found the boy with a man after a sensor in Sunland Park went off Saturday night. The man ran back across the border, leaving the toddler alone.
Sunland Park is a city of about 18,000 people with Ciudad Juarez, Mexico, to the south and El Paso to the east.