Donald Trump has said he expects a post-Brexit trade deal with the UK to be completed "very, very quickly" and that he will be coming to London.
The US President hailed the "very special relationship" he had developed with Theresa May as the pair met for talks on the margins of the G20.
Trump said he expected an agreement on new trading arrangements with Britain to be "very powerful" and would be great for both countries.
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He said: "There is no country that could possibly be closer than our countries. We have been working on a trade deal which will be a very, very big deal, a very powerful deal, great for both countries and I think we will have that done very, very quickly."
He added: "Prime Minister May and I have developed a very special relationship and I think trade will be a very big factor between our two countries."
Trump said he "will be going to London". Asked when, he replied: "We'll work that out."
The Prime Minister later chatted with Trump's daughter Ivanka at the launch of a World Bank project to boost the number of women entrepreneurs.
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In a 50 minute meeting, Trump and May agreed to prioritise work on the trade deal so it can be completed "as soon as possible" after Brexit, a UK Government official said.
They also pledged to look at areas where trade can be deepened before Britain quits the EU. Trump made it clear he believes Britain will "thrive" once it has left the bloc.
The discussions ran 20 minutes over schedule and the leaders continued to chat as they walked to another meeting at the Hamburg summit.
A "significant" amount of time was spent on trade and the discussion was entirely "positive", the official said.
No date will be set out today for Trump to visit the UK. May also held a 20 minute meeting with her Japanese counterpart Shinzo Abe, which focused on trade and North Korea's nuclear missile programme.
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Japan's new trade deal with the EU, signed off on Thursday, "could form the basis" of an agreement between London and Tokyo following Brexit, May told the Prime Minister.
May did not raise President Trump's decision to pull the United States out of the Paris Agreement on tackling climate change during their formal bilateral meeting.
Officials said the PM brought up the issue as they walked between meetings and "said to him face to face that she hopes the US will rejoin" the accord.
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