Carlton Street car park which adjoins the bus station is being wrapped in scaffolding and screens, before work to remove it gets under way.
The Wyvern Theatre car park will also close in mid-August and be demolished by the end of the year.
The redevelopment is the latest stage in the Â£350m Kimmerfields development in the town centre.
The Princes Street multi-storey car park and adjoining apartments have already been built as part of the scheme.
Work to turn Fleming Way into a bus-only route is due to begin in 2016.
It will see the elevated section of the road lowered to pedestrian level and an "uninviting" subway removed.
Once the changes to Fleming Way have been completed the bus station will relocate to the western end of the road.
Developers say the area will include a "linear park" connecting the business district and the station to the north with shops and restaurants.
The project is being run by Forward Swindon, which is leading the regeneration on behalf of Swindon Borough Council.
Forward Swindon is an independent company established and funded by the council in 2010 to "deliver and facilitate economic growth and property development in the town".
Preparation work has started on the second phase of a project to regenerate Swindon town centre.