Bryan McGinty, aged in his 70s, was found dead in his Leabank Court flat in Manurewa on June 24.
A neighbour said paramedics who found Mr McGinty's body told them it was likely he had been dead for five or six days.
Haumaru Housing is runs the flats, which are 51 percent owned by charity The Selwyn Foundation and 49 percent owned by Auckland Council.
Mr Goff said the situation must be fully investigated.
"I have directed council officers to seek a full report from the board of Haumaru on this matter. Should it be required we will seek an independent review of how this matter was handled."
Haumaru Housing chief executive Gabby Clezy said they were deeply saddened by Mr McGinty's death.
"We put the care, concern and dignity of our tenants first and foremost and we are communicating with his family to help them through this difficult time," she said.
"We had been working with Mr McGinty for some time to provide support so he could continue living independently in Leabank Court. We last spoke to him just days before his passing."
Haumaru Housing has more than 1400 senior citizen flats.
