The collision took place on Repington Road, near its junction with Bagot Grove, in the Sneyd Green area of the city at about 17:00 BST on 14 April.
The boy, who was from the area, died in Birmingham Children's Hospital on Tuesday.
Staffordshire Police confirmed it was investigating the incident and appealed for any witnesses to come forward.
More on this and other stories from Stoke and Staffordshire
A seven-year-old boy has died in hospital five days after he was hit by a car in Stoke-on-Trent.