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The position is vacant after chief executive Shimrath Paul left the museum to take up a senior medical centre management job in Indonesia last week.
The chief executive role was one of the country's main museum jobs, Mr Crombie said.
Mr Paul, who had been at Otago Museum for 22 years, had a salary of $310,793, which is believed to be higher than that received by anyone else in New Zealand museum management, including the head of Te Papa.
Mr Crombie said the next chief executive's salary would be partly dependent on the successful candidate, but indicated it was likely to be lower than previously.
An "appropriate package" would be offered for the "right person".
The board had already begun choosing a recruitment consultant to help find a new chief executive.
Subsequent advertising could be over three to four weeks and was likely to be local, national and international, he said.
He hoped to have the position filled by the start of next year.
The museum's management team would handle the interim running of the institution until the next chief executive took over.