Tucson police say they will release the suspect's name at a news conference Friday afternoon.
The attack occurred on Wednesday as the woman walked home after buying groceries in a pedestrian area of south Tucson with a housing development and big box store. The walking path where she was taken was more secluded.
Police said the victim and unharmed baby were taken to a hospital.
Detectives searched the area for hours and sought help from the public. The department released a composite sketch on Thursday afternoon.