Justin King, who ran the supermarket for a decade, said the "last thing" any current supermarket boss would reveal was their intention to put up prices.
But he added it was "very clear" shoppers would face "higher prices, less choice and poorer quality".
The main supermarkets declined to speak to BBC Panorama.
Mr King, who ran Sainsbury's until 2014, said: "Brexit, almost in whatever version it is, will introduce barriers.
"That makes it less efficient which means all three of those benefits - prices, quality and choice - go backwards."
Food and farming combined are the UK's biggest manufacturing sector and the EU is involved all along the chain - from what grows in British fields to the labels in shops.