US President Donald Trump said the US and UK have the “greatest alliance the world has ever known” in a news conference with Theresa May.
The prime minister said they had a “great relationship” that they should “build on”.
But they also “openly” discussed their differences, such as on climate change and business with Huawei, Mrs May said.
Labour leader Jeremy Corbyn urged Mr Trump to “think on about peace” as he addressed protesters in Westminster.
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As the two leaders paid tribute to the “special relationship”, Mr Trump promised a “phenomenal” trade deal after the UK leaves the EU, suggesting it could double or treble trade.
Asked if the NHS could be included in a future trade deal, Mr Trump said “everything is on the table”.
But Mrs May said any agreement between the two countries would follow talks about “what should or should not be in that trade deal for the future”.
Media captionThe US President promises a post-Brexit trade deal with the UK “two or three times as big as now”
Image copyright Reuters
Image caption Mr Trump and Mrs May held a meeting with business leaders on Tuesday morning
Earlier, Mr Trump said to Mrs May, who plans to resign on 7 June: “Stick around. Let’s do this deal.”
BBC assistant political editor Norman Smith said that Mr Trump’s remark to “stick around” was meant almost as a pleasantry rather than a serious plea for Mrs May to reconsider her plan to stand down. “It’s just trying to make this a little bit easier for her,” Mr Smith said.
He added that although Mr Trump “talks a good game” over a trade deal, the UK is likely to be in for a “slow haul”, with most deals taking around 10 years.
Media captionThe US president arrives for talks with Mrs May at Downing Street
Image copyright Reuters
Image caption Mr Trump and the first lady were taken to Downing Street in a limousine known as the Beast
The US president is also set to hold talks with Conservative leadership hopeful Michael Gove, according to sources close to the environment secretary.
Mr Trump praised Mr Gove’s leadership rival Boris Johnson ahead of his visit to the UK.
A source close to Mr Gove told the Press Association “nothing has been fixed yet”, but added: “Mr Gove was asked last night by Mr Trump’s team if he would be able to meet the president today. He said yes.”
Meanwhile, protesters have gathered in central London to voice their opposition to President Trump’s visit. Demonstrations are also planned in Birmingham, Stoke, Sheffield, Glasgow, Edinburgh, Chester, Leicester, Oxford and Exeter.
Police said more than 3,000 officers have been deployed for the president’s visit.
Mr Corbyn – who boycotted the state dinner – was joined at the rally by members of other political parties, including the Liberal Democrats and the Green Party.
He said he had not refused to meet Mr Trump, saying: “I want to have a dialogue.”
But he criticised the US president’s attack on Mayor of London Sadiq Khan. “I am proud that our city has a Muslim mayor, that we can chase down Islamophobia, anti-Semitism, any form of racism within our society,” he said.
Media captionAnti-Trump protesters gathered in Trafalgar Square, as the US president meets the PM
Image copyright EPA/ANDY RAIN
Image caption Protesters against the presidential visit have launched a Trump baby blimp on Parliament Square
Image copyright PA
Image caption Marchers gathered in Trafalgar Square and walked down Whitehall
Mr Khan defended his party leader Mr Corbyn’s decision to boycott the state dinner at the palace.
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Mr Khan said he felt he and Mr Corbyn shared the view that a “close relationship” with the US president was important.
But he added: “What shouldn’t happen is a state banquet and a state visit with the red carpet rolled out. I think it’s inappropriate. I think those visits should be reserved for leaders who have done something and deserve that. I think it sends the wrong message to be seen to condone some of the things this president has said and done.”
At the protest
Image copyright PA
By Marie Jackson, BBC News
The whistling and whooping ramps up as Labour leader Jeremy Corbyn takes to the stage in Whitehall. The rainy streets are now packed with cagoule-clad protesters holding aloft umbrellas and placards saying “Dump Trump”.Some have their faces covered with #trumpstinks masks, others wear badges saying “another nasty woman against Trump”.There are mums with small children in buggies who have given up a day in the park to make their young voices heard, alongside seasoned protesters and US expats.A little further up the street, police officers are dealing with a disturbance by the English Defence League.The rain’s holding off for the moment but Mr Corbyn has a message for the visiting president.
“Think on, please, about a world that is aiming for peace and disarmament, that defeats racism and misogyny.” The crowd roars cheers of support.
On Monday Mr Trump was welcomed by the Queen and had lunch at Buckingham Palace with senior royals. He also visited Westminster Abbey and had tea at Clarence House with the Prince of Wales and the Duchess of Cornwall.
He attended a state banquet at Buckingham Palace in the evening, at which Mr Trump praised the “eternal friendship” between the UK and US.
The Queen said the countries were celebrating an alliance which had ensured the “safety and prosperity of both our peoples for decades”.
The president’s visit coincides with the commemorations for the 75th anniversary of the D-Day landings, which the Queen, Mr Trump and other heads of state will attend at Portsmouth on Wednesday.
Media captionDid the Queen call out President Trump over international institutions?