At a recent Settlers Museum board meeting, the board members favoured closure of the museum until the $35 million project is completed, late next year.
A small part of the museum, comprising the former New Zealand Railways bus station booking hall and a nearby room, has been open this year, offering two small exhibitions.
The rest of the complex has been closed, but staff have continued with redevelopment work.
The council's community development committee further considered the matter at a meeting this week.
Museum director Linda Wigley told the committee the museum complex was largely a construction site, and that some visitors had responded negatively to the limited offerings in the booking hall area.
The committee chairman, Cr Bill Acklin, had earlier commented that the closure was not a "fait accompli" and that the council would also discuss alternatives to closure.
Interviewed after the committee meeting, Cr Acklin said the full council would consider a report from Ms Wigley at its next meeting, on March 28.
Among the eventual possibilities were for the museum to close at the end of April, or to remain open somewhat longer, perhaps with more on display, he said.
If the museum closes to the public, a classroom at the complex will still be used by visiting school pupils and the museum will also maintain a vigorous outreach programme, including activities at the Dunedin Gasworks Museum, officials have said.
