Emergency services were called to Batemoor Road in Jordanthorpe at 1.30pm on Sunday, November 7.
It has been reported that the baby suffered serious injuries after he was hit by a firework.
Family members wrote on the community Facebook page Only in Jordanthorpe that the firework had been ‘thrown’ at the boy’s pram.
The one-year-old suffered burns and was rushed to hospital by ambulance. Police said his injuries are not believed to be life-threatening.
An investigation is now underway. No arrests have been made over the incident.
A spokesperson for South Yorkshire Police said: “Officers have spoken to a number of individuals in connection to the incident and enquiries are ongoing.”
Anyone with information about the incident can call South Yorkshire Police on 101, or anonymously call Crimestoppers on 0800 555 111.