The overture from the US President came during a successful G20 summit for Theresa May, in which she held “positive” free trade talks with leaders of the world’s three biggest economies – the US, China and Japan.
Hailing the talks in Hamburg a success, the Prime Minister said: “Britain has always been a great trading nation and as we leave the European Union we will seize the exciting opportunities to strike deals with old friends and new partners.
“I have held a number of meetings with other world leaders at this summit and have been struck by their strong desire to forge ambitious new bilateral trading relationships with the UK after Brexit.“This is a powerful vote of confidence in British goods, British services and the British economy and we look forward to building on these conversations in the months ahead.”
Confirming plans to visit London, the US President hailed the “very special relationship” he has developed with Mrs May, saying he expected an agreement on new trading arrangements with Britain that was “very powerful” and would be great for both countries.
“There is no country that could possibly be closer than our countries,” he told Mrs “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.
“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.”
Mr Trump said he “will be going to London”. Asked when, he replied: “We’ll work that out.”
Mrs May later said: “We are still working on a date”.Describing the “positive atmosphere” between the two, a senior Government official said: “The President made clear that he believed the UK would thrive outside the European Union.”
The pair agreed to prioritise work on the trade deal so it can be completed “as soon as possible” after Brexit.
They also pledged to look at areas where trade can be deepened before Britain quits the EU.
Mrs May said afterwards: “We are optimistic and positive about this.”International Trade Secretary Liam Fox will begin trade talks with US Secretary of Commerce Wilbur Ross, who was present at the meeting, on July 24.
Mr Trump’s upbeat remarks are in stark contrast to his predecessor, Barack Obama, who warned that Britain would be “at back of the queue” for a trade deal with the US if it voted to leave the EU.
He also told Mrs May he had “a tremendous meeting” with Russian President Vladimir Putin on Friday.
Afterwards Mr Putin said: “TV Trump is very different from the real Trump, from the Trump I had before me.”
Mr Trump’s comments came the day after China’s President Xi Jinping told Mrs May that Brexit could be a global force for good.
The Prime Minister also held a meeting with Japan’s Shinzo Abe yesterday, agreeing that the free trade deal the country has just signed with the EU could form the basis of a post-Brexit pact with Britain. A meeting with Narendra Modi, of India, was also positive.
A Downing Street insider said: “Prime Minister Modi said that he wanted to see the economic relationship between the two countries getting deeper both now and after Brexit.”
Mrs May was forced to insist she will not stand down early, despite intense pressure since losing her Commons majority in last month’s election.
A poll on Friday gave Labour an eight-point lead over the Conservatives and Mrs May acknowledged that the result “didn’t come out as I hoped it would”.
But she added: “There are two ways the Government can react to that. We can be very timid and sit back or we can be bold – and that’s exactly what we are going to be.”
In a speech next week she will set out her continued commitment to social and economic reform.
Asked if she would still be in post for the US leader’s visit, Mrs May said: “I will look forward to welcoming President Trump when he comes to the UK.”
The pair’s talks also covered the ceasefire in Syria which was due to start at 10am British time today, China’s over-production of steel, the Iran nuclear deal, North Korea and the joint action being taken to defeat Islamic State.
They also agreed to work together to remove extremist content from the internet and stopping the flow of money to terrorist groups.
The discussions ran 20 minutes over schedule and the leaders continued to chat as they walked to another meeting, when Mrs May raised the issue of climate change.
The Prime Minister later said she was “dismayed” by America’s withdrawal from the Paris Agreement and said she had urged President Trump to rejoin.
In an extraordinary move, Mr Trump’s daughter Ivanka sat in for him at a round-table meeting with world leaders.
Wearing an elegant pink dress with bow sleeves, the 35-year-old mother-of-three briefly filled in for her father while he held bilateral meetings with other leaders, sitting next to Mrs May and German Chancellor Angela Merkel.
A White House spokesman said: “Ivanka was sitting in the back and then briefly joined the main table when the President had to step out.”