The video, which shows the aftermath of the collision in High Green, has already been seen more than 11,000 times on video sharing website YouTube.
The 63-second clip shows a silver vehicle scattering debris across the carriageway before seeming to come to a sudden halt further down the road.
Children can be heard shouting while a female voice can be heard reassuring the youngsters "alright everybody?" A voice can also be heard exclaiming "Oh my God" during the clip.
It is not clear exactly where and when in High Green the footage was filmed but it was uploaded to the site yesterday.
The video, entitled "Caught on Dash Cam Idiot Driver Speeding Crashes Into The Back of bus Carrying Children Sheffield UK" is one of dozens on a website called IdiotUKDriversExposed.
The clip begins with film shot from the dashboard of the bus as it travels along a suburban road. A vehicle can be seen coming in the opposite direction before a loud bang is here and then a silver vehicle is seen passing on the driver's side in a haze of dust and shattered car parts.
The vehicle appears to narrowly miss a dark coloured van and then can be seen coming to a stop further down the road.
People can then be seen walking in front of the bus windscreen with other cars also arriving in the aftermath of the collision.
Commenting on the video on YouTube, user Katie Holmes said: "This happened outside my home and my little girls and all her friends were on that coach. returning from a singing concert.
"When I got to the coach my daughter was mid asthma attack scary as hell. Teachers were amazing though."