Hull City Council is considering cutting the role of chief executive after failing to fill the £160,000-a-year post.
The position was advertised in July but no suitable candidate was found, despite two people being interviewed for the job.
The vacancy arose when former chief executive Nicola Yates resigned in July 2012.
The council said a decision would be made at a meeting in October.
Since Ms Yates' departure the job has been filled on a temporary basis by Darryl Stephenson, who has held senior positions at Hull and East Riding of Yorkshire Council.
Stephen Brady, Labour leader of Hull City Council, said: "The appointment panel will reconvene in early October to decide the best way forward in relation to the post of chief executive of the council."
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