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The proposal from New Zealand's largest insurer comes as it also plans to introduce risk-based pricing, which could mean customers who live in areas prone to natural disasters and severe weather pay more for home insurance, in a progressive roll-out from July 1.
Seven branches, including South Dunedin and Alexandra, may close, which would reduce AMI's network to 51.
Seven staff across the two branches would be affected by the proposal.
Central Otago deputy mayor Neil Gillespie said he hoped any decision AMI made considered those people who would be adversely affected.
''This is becoming an unfortunate and unwelcome trend in small-town New Zealand,'' he said.
''I understand the business sense.
''But Central Otago, for example, is isolated and has a high population of elderly who rely on those branches. I hope AMI think about those things because that, too, could affect their business.''
Alex Geale, executive general manager of AMI's retail network, said it was reviewing its network as a result of customer preferences changing.
''We are increasingly finding that our customers across New Zealand are preferring to contact AMI through other means, rather than visiting their local stores.''
Ms Geale confirmed the insurer had ''tens of thousands'' of customers in Otago and Southland.
Branches are located in Dunedin, Gore, Invercargill, Mosgiel, Oamaru and Queenstown in addition to those in Alexandra and South Dunedin, which were ''under review''.
Ms Geale confirmed an 18% increase in customers using messaging apps such as Facebook messenger, WhatsApp, WeChat and AMI's artificial intelligence-powered ''chatbot'' Amy over the past 12 months, which had become the ''customer service norm''.
AMI had also noted a 35% increase over the past year in customers using the online payment system.
''While it is disappointing that we might have to close a small number of stores, AMI will continue to be there for New Zealanders, whether they live in urban areas or smaller communities, by providing service in the way they want it to be - conveniently, efficiently and effectively.''