The force received a report at around 1pm on Monday, July 19, that the young girl had been hit on Dumbarton Road, near the Scotfresh store.
Emergency services attended and the four-year-old was taken by ambulance to the Royal Hospital for Children in Glasgow where she still currently remains receiving treatment for a serious leg injury.
The car did not stop at the scene and Police Scotland has confirmed that inquiries are ongoing to try to trace the driver to establish the full circumstances around the incident.
Officers are appealing for anyone who was in the area at the time of the incident to come forward with any information they may have, and are particularly keen to hear from drivers who have dashcam footage.
Sergeant Roy McCarney of the Road Policing Unit said: “A young child has been left with a serious leg injury as a result of this incident and it is absolutely vital that we trace the person driving the car involved.
"The surrounding area would have been busy with pedestrian and vehicle traffic at the time and I am appealing for any witnesses, in particular any motorists who may have dashcam footage of the incident, to come forward to assist the investigation.
“Anyone with information is asked to contact Police Scotland through 101, quoting incident number 1550 of Monday, July 19."