Adrian Chiles's decision to move from the BBC to ITV earlier this week has left his role on the reality series spinoff open.
Balding, who has appeared as a panellist on the show in previous series, has been mentioned as a potential new host, according to The Guardian.
BBC business editor Robert Peston has been linked to the post by The Sun, while Evan Davis, Chiles's former Working Lunch co-presenter Adam Shaw, newsreader Sophie Raworth and BBC economics editor Stephanie Flanders have also been tipped as possible replacements.
Showbiz presenters linked to the role include Big Brother's Big Mouth host George Lamb and Radio 1's Matt Edmondson. Lamb's participation would be dependent on when the sixth series eventually airs, because of his Big Brother commitments.
Fans of the programme on the DS forums have also suggested that former contestants such as Raef Bjayou, Saira Khan or Ruth Badger could helm the series.
The Apprentice: You're Fired, which has been presented by Chiles since 2006, moved from BBC Three to BBC Two in 2007.