A complaint alleging bullying and intimidation and raising concerns about Ms Graham’s behaviour towards some staff was sent to Dunedin Mayor Jules Radich in April, the Otago Daily Times can reveal.
The complainant continues to work at the city council.
The ODT asked Ms Graham in May about perceived concerns relating to her management style, as well as seeking clarity about other subjects, including the complainant’s employment status.
"There is no way to respond to the questions you are asking without infringing on the privacy rights of our staff", Ms Graham responded.
"I can confirm we follow up all matters of concern, when raised, appropriately."
Mr Radich said he had no comment to make because of legal privilege, privacy and fairness.
The ODT asked Mr Radich earlier this month about bullying claims.
"As has been said repeatedly now, I and my councillors have legal obligations to protect privacy and legal privilege", he said.
"It is vital to ensure fairness and prevent prejudice, so I will not be commenting on any specific issues."