The crash happened between the junctions for Rowley Lane and Woodsome Road.
One person was also taken to hospital following the accident on Friday morning.
A spokeswoman for West Yorkshire Police said: “We were called at 10.50am to Penistone Road near the junction with Woodsome Road following reports of a collision involving a bus and a Skoda estate car.
“One person was arrested. Another person was taken to hospital. Their injuries are not thought to be serious.”
An eyewitness who drove past the scene said: “The bus looked very badly damaged and there were loads of police and ambulances at the scene.
“Lots of drivers were held up around the bottom of Rowley Lane.”