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Sue Bidrose was appointed to the role in November 2013 and her position has to be advertised this year.
Dr Bidrose declined to comment on whether she sought to extend her time in the role.
However, the council’s chief executive appointment committee will meet tomorrow from noon.
The public will be excluded.
The members of the committee are Mayor Aaron Hawkins and Crs David Benson-Pope, Mike Lord, Marie Laufiso, Jim O’Malley and Chris Staynes.
Dr Bidrose was the general manager of services and development for the council before she accepted the top job, replacing Paul Orders.
Dr Bidrose was reappointed for a two-year term in late 2018.
She has consistently sought to decline or limit pay rises for herself while she has been chief executive.
She started on $350,000 and in August last year she was reported to be on $444,000, after being obliged to accept more pay.