The seven-year-old and his brother had been standing on a traffic island in Gilmerton Road when a car clipped him and kept driving.
Chief Inspector Alwyn Bell said: “It may well be the driver is not aware that anything happened so we would appeal for anyone who may have been driving in Gilmerton Road this lunchtime, and who may have seen the boys on the traffic island, to get in touch. We’re also keen to hear from anyone who may have seen the vehicle involved to contact us on 0131-311 3131.”
The incident, which involved a silver or gold car, took place at around 12.40pm. No further details are available.