Teitur Gudmundurson was travelling on the 25 Stagecoach service through Lowedges at around 7pm on Friday evening when he noticed a car hurtling down the road at speed.
"I saw it fast approaching and could tell already that it wasn't going to be able to manage the turn at the speed it was travelling at," said Teitur, of Celandine Court, Bradway.
The bus was travelling on Lowedges Road, when the car being pursued emerged from Gresley Road and crashed into it.
Teitur says he was sat near to where the car struck the bus.
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The 44-year-old said: "I think I yelled out in shock. I think I might have sworn.
"We felt the impact as it hit.
"I looked out the front window and the car drove off.
"The bus came to a halt at the next stop which was a very short drive forwards.
"But as we exited, someone else yelled and we could see there a police car that was pursuing it so it seemed a chase had been going on."
No-one is believed to have been injured in the collision, but a number of panels on the bus fell off following it.
The emergency services were called following the incident, and police are believed to have visited the scene in a different patrol car a short time later.
South Yorkshire Police have been contacted for comment.