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UPDATE 1-Trump says UK health service should be part of U.S. trade talks

(Adds health minister, Labour leader, background)

LONDON, June 4 (Reuters) - U.S. President Donald Trump said Britain's public health service should be on the table in future trade talks between the two countries after Brexit, prompting Prime Minister Theresa May to say some areas of the economy might be off-limits.

The National Health Service (NHS) is a cherished institution for many Britons. Created after World War Two, it provides a wide range of free services ranging from routine consultations to life-saving operations.

"I think everything with a trade deal is on the table," Trump told reporters during a visit to London. "So NHS or anything else, or a lot more than that. But everything will be on the table, absolutely."

May, speaking at the same news conference , quickly followed up: "But the point about making trade deals of course is that both sides negotiate and come to an agreement about what should or should not be in that trade deal for the future."

Britain's health minister, Matt Hancock, also rebuffed Trump's proposal.

"Dear Mr President. The NHS isnt on the table in trade talks - and never will be. Not on my watch," Hancock, who is a contender to replace May as prime minister, said on Twitter.

The opposition Labour Party has focused on fears among voters that the NHS might be privatised as it tries to capitalise on the Brexit crisis within May's Conservative Party.

"Theresa May stood next to zrealDonaldTrump as he said the NHS will be 'on the table' in a US trade deal. And thats what Tory leadership contenders and (Brexit Party leader Nigel) Farage are lining up for the No-Deal disaster capitalism plans they have," Labour leader Jeremy Corbyn said on Twitter.

"They all need to understand: our NHS is not for sale."

(Reporting by UK bureau Writing by William Schomberg; editing by Stephen Addison)