It is the second high-profile departure from The Sun this year, with Roland Agambar, head of marketing, moving to Express Newspapers in March.
Canning is described as the mastermind of the price wars that have helped push The Sun's circulation to around 3.7m, while tabloid rival the Daily Mirror languishes around the 2.1m mark.
In an interview last week, David Yelland, editor of The Sun, declared his newspaper had scored a "total victory" over the Mirror, which has paid heavily for slashing its cover price to 20p for a period.
Canning joined The Sun in 2000 from KP Foods, where he was marketing director. He has also worked for Heinz and Terry's.
Janet Marsden, human resources director at the Big Food Group, said: "Nick joins at an important time in our recovery strategy. His background and experience equip him to make a big contribution to our future success."
The Big Food Group, which also owns the cash-and-carry chain Booker, has been the subject of takeover rumours and management buyout stories. Yesterday, the Icelandic company Baugur has bought a 14.9% stake in the company.
Iceland attempted to reposition itself as a champion of organic food, but found that its strategy was not paying off in increased sales. It has now switched back to its roots of offering value-for-money frozen foods, and its advertising is heavily price led.
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