The incident happened at around 1pm on Monday, April 12, at the junction of Kilmarnock Road and Langside Avenue.
An eyewitness told The Scotsman that a car “suddenly skidded to a stop” in the middle of the road, before a man jumped out and ran towards Queens Park.
She said: “I was just crossing the busy junction at Kilmarnock road near the Corona pub on a run and suddenly this car skidded to a stop in the middle of the road blocking one of the lanes on the Langside area.
"The male driver then ran out of the car and headed toward Queens Park. He must have clocked that the police were following him as two police cars pulled to a stop around the vehicle, which had damage to the front left hand side of it.”
She continued to say that one police man began directing traffic following the incident and that everyone around was “shocked and confused”.
She added: “I continued my run into the park and noticed a police car near the pond and then I noticed a police van leaving from the entrance beside the New Victoria hospital. The man was definitely on the run from the police.”
A police spokesperson said: “At around 1.10pm today a car failed to stop for officers on Pollokshaws Road, Shawlands.
“One man has been arrested in connection with the incident and enquiries are ongoing.
“Anyone with information is asked to contact police via 101 quoting incident number 1429 of April 12.”