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The boss of Tesco today braced customers for price rises in the wake of the pound’s slump after the Brexit vote.

John Allan, the supermarket’s chairman, said it was “very likely” that the cost of some products would go up.

Sterling was trading at about $1.22 this morning, some 17 per cent lower than before the shock June 23 referendum result, and has dropped sharply against the euro.

The decline will almost inevitably push up the cost of imported food and ingredients from overseas for products made in Britain.

“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,” Mr Allan told BBC radio.

He believes inflation may “nudge up” two or three per cent, with food prices contributing to this — but dismissed suggestions food prices could rise on average by between 10 and 20 per cent, insisting it would be “much less”.

Tesco was seen to have won the “Marmitegate” battle with Unilever after it sought to impose a 10 per cent hike in the cost of products supplied to the supermarket giant.

Mr Allan, also chairman of business group London First, declined to comment on the bust-up but appeared to suggest it had not been such a clear-cut rejection, as portrayed, of any price rises.

He added: “We see our role as retailers not to prevent our suppliers from making legitimate cost increases where they think that’s necessary but certainly scrutinising those cost increases very hard.”

The supermarket chief also warned Theresa May against a draconian clampdown on immigration which could harm the British economy.

He said: “Industries like the agriculture industry, picking and packing fruit and veg and meat, are heavily dependent on migrant workers, not just seasonal migrant workers but people who are here 52 weeks a year.

Our fresh food suppliers are very concerned that is properly taken into account.”

Earlier this month, sterling plunged after the Prime Minister appeared to suggest she was prioritising immigration control over economic concerns.

But yesterday she told MPs the UK can curb immigration and get a good trade deal with the EU. She said: “We are going to be ambitious for what we obtain for the United Kingdom and that means a good trade deal as well as control of immigration.”

Meanwhile, French ambassador Sylvie Bermann said “of course” Britons in France would be allowed to remain after Brexit if the same applied to French nationals here.

She said: “I think it will be dealt with on a reciprocal basis because there [are] 300,000 French citizens in this country, almost the same as UK citizens in France.”