King, who was head of the supermarket for 10 years, told the BBC Panorama consumers were “completely in the dark” about the effect Brexit will have on their weekly shop.
He said the UK’s vote to leave the European Union last June had pushed the pound lower and food prices higher.
Should the UK fail to reach a free trade deal with the EU in its Brexit negotiations, food inflation is expected to soar.
King said Britain faces losing the support that the EU membership provides retailers in finding the best suppliers and markets throughout Europe.
He said the EU had lifted standards and allow the UK to get its hands on out-of-season vegetables all year round.
"Brexit, almost in whatever version it is, will introduce barriers,” King said.
"That makes it less efficient which means all three of those benefits - prices, quality and choice - go backwards."
Vote Leave campaigner John Mills blames EU for keeping food prices high
However, Brexiteer and manufacturing boss, John Mills, said he believes the EU has kept pricing artificially high.
Mills said tariffs had kept food prices up inside the EU.
"Food prices inside the EU vary from food product to food product, but the average is something like 20% higher than they are in the rest of the world - so there is very substantial scope for food prices coming down if we switch sources of supply outside the EU."
He also claimed that institutional arrangements meant food prices were left much higher to increase farmers' incomes.
British farmers fear Brexit will impact business
British farmers are worried that a bad deal on Brexit could hurt their business.
John Davies, a livestock farmer from Powys in Wales, told the BBC: “I'm really scared of imports - produced to completely different standards [with] hormones, you know, feedlot beef, you know.
“We're based on green and pleasant land, high environmental standards. We really are proud of that."