Police and paramedics were called to reports of a child unconscious at an address in Acocks Green on Thursday morning.
The boy was taken to hospital but West Midlands Ambulance Service said “nothing could be done to save him”. He was later pronounced dead.
A 44-year-old woman remains in custody and is being interviewed by detectives, West Midlands Police said.
The force is treating the boy’s death as unexplained and a post-mortem examination is due to take place.
Detective inspector Joe Davenport said: “This is a truly tragic incident in which a young boy has been found dead. We understand he had some underlying health issues and was admitted to hospital recently.
“We’ve launched an inquiry to understand what happened in the days leading up to his death and to determine if any factors have contributed to his death.”
An ambulance service spokesman said: “Two ambulances, a paramedic officer and the Midlands Air Ambulance Critical Care Car were sent to the scene. On arrival we discovered one patient, a boy, who was in a critical condition. He received specialist trauma care at the scene before being transported to Heartlands Hospital.
“Sadly, it later became apparent at hospital that nothing could be done to save him and he was confirmed deceased.”