The bird has been spotted in Aberdeen Rd, Earls Rd, Lock St, Jacobs Ladder, Valpy St and Norfolk St in St Clair, as well as in neighbouring St Kilda.
Dunedin city councillor Conrad Stedman said he was also chased by the turkey on Jacobs Ladder.
"I had to duck out of the way. It was trying to take me out. It's a monster of a bird."
A St Clair resident said he noticed the turkey a month ago.
The "really big bird" would not fit in his oven, so the resident had made attempts on social media to find someone willing to give it a home.
City council environmental health and animal services manager Ros MacGill said the bird did not come under council control because no reports of disruption or nuisance had been received.