British shoppers should brace themselves for higher prices following the steep fall in the pound since the Brexit vote, the former boss of Sainsbury’s has said.

Justin King said supermarkets would be unable to absorb the recent rise in the cost of importing goods caused by the falling value of the pound against the dollar and euro. Instead, said King, they would soon pass on the costs to consumers in higher prices.

Mike Rake, the chairman of BT, speaking at the same conference as King in London on Wednesday, said imported mobile phones and broadband home hubs were already 10% more expensive and the cost would have to be passed on to consumers in the near future.

Sterling has come under intense pressure over the past week on signals from Theresa May’s government that it will go for a hard Brexit deal that sacrifices single market access in favour of stricter immigration controls.

Since the vote to leave the European Union the pound has fallen 20% against the dollar and 22% against the euro. It was trading at just under $1.22 on Wednesday afternoon, after a further plunge in value on Tuesday.

Business lobby groups have previously said consumers will face a steep rise in import costs on a range of goods from cars to clothes as a result of the currency drop.

But the UK’s annual inflation rate stayed at 0.6% last month, confounding City forecasts of a small rise in the cost of living due to imports becoming more expensive.

King, who is now a vice-chairman of private equity firm Terra Firma, said: “Retailers’ margins are already squeezed. So there is no room to absorb input price pressures and costs will need to be passed on.

“But no one wants to be the first to break cover. No business wants to be the first to blame Brexit for a rise in prices. But once someone does, there will be a flood of companies because they will all be suffering.”

Rake, a former chairman of the accountancy firm KPMG, said: “We have already had a 10% increase in mobile phone handsets and home hubs, and that will need to be passed on at some point.”

It is believed that many high street retailers and consumer goods importers have hedged their currency risks for at least the rest of the year. But that only means increases in the cost of some imported goods being delayed until next year.