Brexit is threatening to sink London's insurance market
John Nelson said that London's position as a global hub of specialist insurance was under threat, with the need for Lloyd's and others to remain in the single market greater than the need to remain in the capital.
Lloyd's could move some of its market from London before Brexit negotiations are complete unless Prime Minister Theresa May indicates a "clear direction of travel" for the UK's departure from the EU, he said.
"Lloyd's is Lloyd's, it's not Lloyd's of London," Nelson told the BBC Today programme on Monday. "If there is uncertainty for a prolonged period of time, then the industry will vote with its feet.
"That would mean moving business, or leaving London, more quickly than the renegotiation timetable," he said.
Lloyd's is the oldest insurance marketplace in the world, having been founded in 1688 as a maritime insurance market.
It was a vocal opponent of Brexit in the run-up to the EU referendum.
Speaking in February, Nelson said that EU membership was a "crucial element" of London's ability to retain its position as a global hub for insurance, citing the importance of "passporting" rights between the 27 EU member states.
Passporting allows insurance firms and banks based in the UK to operate freely across the what is known as the European Economic Area (EEA), which is made up of the EU and certain other countries. including Norway.
Last year 11% - £2.9 billion - of Lloyd's transactions came through the EEA.
Nelson's comments come as global leaders use the annual G20 summit express concern at the possible implications of Brexit.
Over the weekend Japan issued a "warning note" to the UK, detailing the potential damage of Britain's decision to leave the EU.
At the G20 meeting, Theresa May was photographed standing on the edge of a "family photo" of world leaders, appearing alienated from key players such as Barack Obama and German chancellor Angela Merkel.
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