The 16th century, three bedroom house at 42 Michaelgate is a tourist attraction and appears on local postcards. The£40 a week property became vacant when Frank Bradford, tenant for the past 28 years, died last month.
Housing director John Bibby said the council, which has 2,000 people on its housing waiting list, had received a dozen written applications for the tenancy and numerous telephone enquiries.
Mr Bibby said the house's internal layout and winding staircase meant it would only be suitable for a family with older children. But he said some councillors had suggested converting it into a tourist attraction.
The council's solicitors believe that because the house was gifted to the city by local clergyman Canon Norman Harding, the council does not actually hold title to the property and so cannot sell it.