The incident happened at around 3pm today, on Pollokshaws Road.
Witnesses later reported seeing a heavy police presence, including armed officers at the Aldi supermarket on Greenlaw Way in Newton Mearns.
A Police Scotland spokesperson said: "Police were called to Pollokshaws Road in Glasgow around 3pm on Tuesday, 13 April, following a report that a car had mounted the pavement before driving from the scene.
"This car has been traced in Greenlaw Way and officers are carrying out searches in the area for the occupants.
"Enquiries into the full circumstances of the incident are ongoing and there is not believed to be any threat to the wider public."