The victim, who is 13 years old, has reported a number of occasions where she has been followed by a man driving a vehicle.
Although the suspect has not approached the girl or hurt her in anyway, this behaviour has made her feel frightened and threatened.
The incidents have happened in various locations, including Penhill, Pinehurst, Moredon, Rodbourne and Cheney Manor.
The suspect, who is elderly, drives a small silver car.
Local Crime Investigator Hayley Beales, from Swindon Community Policing Team, said: “While we have made an arrest as part of our investigation, we need to speak to any potential witnesses who may be able to assist us.
“This young girl has been made to feel very afraid by the behaviour of this man and this has been a repeated pattern of behaviour over a period of time.
“If you believe you can assist with our enquiries then please call us.”
An 81-year-old man has been arrested on suspicion of stalking and is currently released on conditional bail while our enquiries continue.
Anyone with information should call LCI Beales on 101, quoting crime reference 54200097374.