
The chimney is part of the Dunedin Gasworks Museum which has now been closed for at least a week while inspections are being carried out.
It is directly next door to the Dunedin South Woolworths supermarket.
A Woolworths spokeswoman said it was informed of the situation on Wednesday afternoon, and noted the council was awaiting an engineer’s report.
"We will stay in contact with the council and assess next steps once the report is available."
Dunedin City Council group manager property services Anna Nilsen said in a statement the cracks were discovered last week.
“A member of the public first noticed a lightning rod running up the chimney was no longer flush with the brickwork.

It was suspected the cracks may have been caused by last month’s 6.8 magnitude earthquake off the coast of Southland.
The site had been closed "as a precautionary measure" while further investigations were done.
“The closure will be for an initial period of one week, while we assess the situation and consider options, but may need to be extended.
“This also has the potential to impact surrounding businesses, and we’re working proactively to keep everyone informed."
An engineer’s report on the structure was expected to be ready shortly.
The Dunedin Gasworks Museum is owned by the DCC and operated by the Dunedin Gasworks Museum Trust. - APL