It's believed the Mitsubishi was towing the Golf when it was stopped by members of the public.
Officers were called and a 52-year-old man from Malmesbury and a 41-year-old man from Swindon were arrested on suspicion of theft of a motor vehicle.
Sgt Katharine Smith said: "We would like to speak to anyone who may have seen the brown truck towing the white Golf around this time on Saturday afternoon.
"It was a sunny day in the centre of the village and lots of people would have been out so I believe someone would have seen something.
"We would especially like to hear from anyone who saw the vehicles being driven."