However, the couple - Michael Beazley and Deborah Wai Kapohe - say their plea for the council to buy their home, to allow them to relocate, has already been rejected.
The couple met Mr Cull and Dunedin City Council staff in the mayor's office on Wednesday afternoon to discuss concerns about their home at 36 Leithton Close, Glenleith.
They said the council erred in signing off on a code of compliance certificate for the home, confirming it conformed with New Zealand Building Code requirements.
The couple checked the certificate before buying the $550,000 property late last year.
It was only days after moving in the couple said they discovered weathertightness faults including leaks, cracks and toxic mould.
Ms Wai Kapohe said Wednesday's meeting was "productive" and "a small step forward".
Among the steps to emerge, council staff had been asked by Mr Cull to respond to the couple's questions, she said.
The couple had previously written to the council asking questions about their home's code of compliance certificate, but received a letter from council staff declining to respond directly to the questions.
Ms Wai Kapohe said she and her husband had been asked at the meeting to prepare a document explaining what was in their home's building file, and what was missing, and outlining their concerns for staff to follow-up.
"I thought that's a great start."
Council city environment general manager Tony Avery and development services manager Kevin Thompson also attended yesterday's meeting, but not chief building control officer Neil McLeod.
Mr Avery said the meeting was "productive" and "gave both sides a better understanding of each party's situation, key issues and a way of trying to work forward on resolving those as quickly as possible".
Staff would work "as quickly as we can" to respond to the couple's concerns once they got the document they were preparing, plus work through the Weathertight Homes Resolution Service (WHRS) or other process, he said.
The parties had also agreed "to deal with things in an open manner and to keep talking about things", he said.
Ms Wai Kapohe said a request for the council to buy their home, or pay for a new rental "safe house" while their own home was repaired, were ruled out when raised again at Wednesday's meeting.
Mr Avery said there were other parties involved in the dispute, and the degree to which the council was liable was "still to be worked through".
"The council doesn't feel that it's responsible to the extent that it needs to purchase the property outright.
"That's got to be worked through the Weathertight Homes Resolution Service."
The couple were still waiting for a full WHRS report, which would determine whether their claim - lodged earlier this year - had been accepted.
That could lead to a financial settlement through mediation or a tribunal, but the couple have also threatened to pursue the council privately through the courts.
