"Alice herself has implied to me, very strongly, that an exit vote would see us sleepwalking to disaster".
Deutsche Bank also revealed that it is reviewing whether to move swathes of its 9,000 UK staff out of the country in the event of an exit vote.
A report from trade body Abta, due on 15 March, is widely expected to go into detail about the consequences of an exit vote.
May wanted to tackle the underlying concerns that led to the E.U. exit vote, rather than simply deal with its symptoms, it appeared.
This would mean Britain Stronger in Europe would be at a distinct financial and publicity advantage, though it could add to the anti-establishment allure of an exit vote.
He said an exit vote "will fire the starting gun on a race to the bottom: a Britain of low wages, unsafe workplaces and a bonfire of workers' rights".
The Bank of England is likely to enter the market to provide liquidity – making sure banks have enough cash – if an exit vote prompts market mayhem and investors attempt to pull cash out of the UK.
On the consequences of the build-up to an exit vote, he added: "Uncertainty is never good for business, although Vodafone is very large and we are in many countries inside and out of Europe so we can deal with uncertainty – it's never good, but we can deal with it".
"If we are to avoid an exit vote in any referendum," the group's president, Roger Carr, told the confederation's annual conference in London, "it is essential that the voice of British business is loud and clear in extolling the virtues of future engagement — not as a reluctant participant but as the linchpin of our wider global trade ambitions".
He told the Financial Times newspaper that an exit vote could end a bilateral deal allowing the UK to vet new arrivals on French territory.
