* Trump says "no deal" Brexit should be an option
* Says could work very quickly on a U.S.-UK trade deal
* U.S. ambassador says would include agriculture, healthcare (Wries through, adds U.S. ambassador on trade deal)
LONDON, June 2 (Reuters) - U.S. President Donald Trump said Britain should refuse to pay a$50 billion European Union divorce bill and "walk away" from Brexit talks if Brussels does not give ground.
Trump told the Sunday Times newspaper ahead of a state visit to Britain, which starts on Monday, that Britain's next leader should send arch-Brexiteer Nigel Farage to conduct EU talks.
Once Britain leaves the EU, which Trump said must happen this year, then he would go "all out" to agree a trade deal.
"They've got to get it done," he said in the Sunday Times interview. "They have got to get the deal closed."
British Prime Minister Theresa May will step down shortly after this week's Trump visit, having failed to win backing for the Brexit divorce deal she negotiated with the EU.
Trump said her successor should pursue a "no-deal" Brexit if he or she could not get more concessions from Europe by the end of October, when Britain is due to leave.
"If they don't get what they want, I would walk away," he said. "If you don't get a fair deal, you walk away."
The 13 candidates already in the leadership race are split between those willing to accept a "no-deal" and those opposed.
In the "no deal" camp are former foreign secretary Boris Johnson, whom Trump praised in an interview with the Sun newspaper on Friday, along with former Brexit minister Dominic Raab and interior minister Sajid Javid.
Trump said the United States could work "very, very quickly" on a trade deal if Britain was not constrained by a transition period agreed with Brussels.
ALL ON THE TABLE
The U.S. Ambassador to Britain, Woody Johnson, said any such trade deal would include agriculture and healthcare.
"In a trade deal, all things that are traded will be on the table," he told the BBC on Sunday. Asked if that included healthcare, he replied: "I would think so".
Concerns have been raised in Britain about accepting U.S. agricultural standards, notably chlorine-washed chicken, and about opening up its state-funded healthcare system to U.S. companies as the price of a trade deal.
"American products would come over, and be allowed to come over," Johnson told the BBC's Andrew Marr Show. "You give the British people a choice, if they like it they can buy it, if they don't want it, they do not have to buy it."
Trump said it was a mistake for the Conservatives not to involve Farage, the Brexit Party leader, in negotiations with Brussels after his success in European Parliament elections.
"I like Nigel a lot. He has a lot to offer - he is a very smart person," Trump said. "They won't bring him in but think how well they would do if they did. They just haven't figured that out yet."
Farage, who led the unofficial campaign to leave the EU in the 2016 Brexit referendum, wants to leave the bloc without any agreement.
His new Brexit party swept to victory in the United Kingdom's European parliament election last month, prompting him to demand a seat at Brexit negotiations.
However, none of the candidates seeking to replace May are expected to offer an olive branch to a longstanding rival who has the potential to split the right-wing vote in Britain.
Trump also said he would have "to know" Labour opposition leader Jeremy Corbyn before authorising U.S. intelligence to share its most sensitive secrets with a hard-left government.
He said Britain must be careful not to jeopardize intelligence-sharing by letting Chinese firm Huawei Technologies Co Ltd into its 5G mobile phone network. ($1 = 0.7910 pounds)
(Reporting by Stephen Addison and Paul Sandle Editing by Chris Reese and Alexander Smith)