Labour councillor Karen Lee, who represents the area, described the six proposed affordable homes as “a travesty” and “shockingly low”.
Local resident Susan Nock has been successful in getting legal orders to protect several trees on the site.
She said: “Why should the residents of Lincoln be made to pay for this by losing social housing and green spaces?”
If the application is given the green light, further details such as the size and layout of homes would need to be agreed in a future application.
The spokesperson for the cathedral said conservation of the building was of the highest priority and received no funding from the government or the Church of England.
“Proceeds from the sale of the land would go to support this work, although at the present time it is not possible to say exactly which project that will be,” they added.
“That will depend on the date of the sale and the other sources of funding for the various works projects.”