He is said to be tall and of normal to slim build with dark brown hair and a short dark beard.
The man was wearing a dark blue cap, a red short-sleeved t-shirt with blue block letters on the front and wearing either light blue knee length shorts or dark blue denim jeans.
In the second incident two girls, also 12, were walking along the western side of Hillsborough Road, opposite the Hillsborough Cemetery.
A man, who came out from the bushes of the cemetery as they walked between Richardson Rd and Goodall St, walked at the same pace and kept directly across the road from them.
He stopped a short time later, pulled his pants down and exposed himself.
The girls then ran to one of their home addresses.
The man is described as being Caucasian, in his late 20s, with dark brown hair and stubble on his face.
He was described as being of slim build, wearing a dark blue cap with a dark grey t-shirt.
Detective Senior Sergeant Marcia Murray said there is a strong possibility that these two incidents were linked.
But did not believe they were related to a report on Monday where a nine-year-old girl was approached by a male and offered a ride.
"We are now asking for members of the community who may have information which can assist police to contact us," Murray said.
"It may be something minor that you have seen but we want to hear about it."
Police patrols around the area have been increased.
Anyone with information is being asked to contact Avondale Police on 09 820 5784 or provide information anonymously to Crimestoppers on 0800 555 111.