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This comes as a group of social agencies, including the Salvation Army and Presbyterian Support, agreed to a statement saying the situation was reaching or had reached "crisis point".
The group said rising rents were making it hard and sometimes impossible for people on low incomes to find affordable rental properties.
"We are seeing a trend of landlords ending and not renewing leases, which forces tenants into a rental market they often cannot afford."
Waiting lists for social housing were growing and more families were living in cars and garages or being put up in motels while they waited for social housing.
The group, led by Dunedin South MP Clare Curran, called on the council to co-ordinate a city meeting focused on identifying the problems and finding short-term solutions.
"We believe the Dunedin City Council can play a strong role given it provides social housing and that housing quality and availability is an objective of its social wellbeing strategy."
They also believed the Government was not doing enough to remedy the problem and that it should be involved in finding a local solution to the problem.
In a statement provided to the Otago Daily Times Mr Cull said it was not the council’s place to lead discussions, but it would be happy to take part in Government-led discussions.
"Through our social wellbeing strategy, the council is already committed to supporting good housing outcomes.
"One of our focuses in recent years has been on supporting the Cosy Homes Trust to increase the number of warm homes throughout the city."
As a provider of council community housing it was committed to matching applicants, particularly those over 55 with low incomes, to suitable, long-term accommodation.
"We will soon be embarking on a social housing plan that looks at how we best manage this into the future, and we’ll be engaging with the appropriate social service organisations as we develop that plan.
"However, the provision of short-term and emergency housing is a core function of central government, not local government," Mr Cull said.
"Therefore we would be happy to participate in central government-led discussions around the current housing situation in Dunedin to understand from them what the short- and long-term solutions might be."
Ms Curran was disappointed the council declined the request, but said she was happy to keep talking to the council about the issue.
She did not believe it should be left to the Government to lead discussions.Instead she would work with Dunedin’s social agencies to find solutions.
"We’ve got to do something now."