Tesco chairman warns that UK food prices are 'very likely' to go up as a result of Brexit

Asked about whether Brexit would increase the price of food, Mr Allan told BBC Radio Four's Today programme: "I think it's probable, it obviously depends on currency, where the pound settles. If it stays where it is, it is likely."

Mr Allan insisted he wanted to protect customers as much as possible after Tesco had a high-profile dispute with a supplier who tried to use the fall in the pound to increase prices.

"Everyone who wants to increase prices will get scrutiny. We are trying to defend our customers from unjustified price increases

"But, that it is likely there will be some price increases going forward - I think is very likely.

"Remember at the moment we have got virtually nil inflation. Inflation I think could nudge up to 2% or 3%, food prices would be a part of that," he said.

Mr Allan warned that suppliers are "very concerned" about restrictions on migrant labour after Brexit, as he insisted Britain could not just take highly skilled immigrants.

"We have about a million, for example, EU citizens living in London, and many hundreds of thousands elsewhere in the country.

"And industries like the agricultural industry, picking and packing fruit and veg, and meat, and so on, are heavily dependent on migrant workers - not just seasonal migrant workers, but people who are here 52 weeks a year.

"And I know our fresh food suppliers are very concerned that is properly taken into account.

"We don't just need brain surgeons and architects, I think we need many people who do much more ordinary, but nonetheless extremely important, jobs," he said.

France's ambassador to the UK, Sylvie Bermann, said she expected French people to be treated the same way in Britain as Britons would be in France after Brexit.

"I think it will be dealt with on a reciprocal basis," she told the BBC.