A mayoral spokesman said Brown has not had any discussions with any of the candidates about the matter and was not considering any council related role post-election.
I have not given consideration at all to any employment for the outgoing mayor
Goff and Brown are long-serving members of the Labour Party and received backing from the party for their respective mayoral bids.
Brown announced on November 8 last year he was standing down as mayor, conceding a two-year affair with council adviser Bevan Chuang had greatly affected his public image.
Two weeks later, Goff launched his bid for the mayoralty, saying Brown's sex scandal had damaged his credibility and ability to get things done.
The Auckland Mayor has executive powers to appoint staff to the mayoral office and can influence other appointments, such as board members for council-controlled organisations.
Last year, Brown appointed former Waterfront Auckland chairman and former Labour Party president Sir Bob Harvey to be overseas investment champion based in the mayoral office.
Goff's main rival on the centre-right, businesswoman Vic Crone, said she would not give Brown a job if she won the mayoralty.
"Apart from definitely having different priorities, we absolutely need change with a fresh approach and ideas," Crone said.
Goff said people were looking for a clean break from the current administration and he envisaged changing most of the staff in the mayoral office.