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, Iran and China, Mrs May said.
Mr Trump also said he turned down a meeting with Jeremy Corbyn, who addressed protesters in Westminster.
The Labour leader had proposed talks, saying he was ready to engage with the US president on issues such as climate change, threats to peace and the refugee crisis.
“I want to be able to have that dialogue to bring about the better and more peaceful world that we all want to live in,” Mr Corbyn told the crowd of protesters.
When asked about Mr Corbyn, the US president said he did “not know him, never met him, never spoke to him”, but added that he had turned down an opportunity to meet the Labour leader on a different occasion.
Mr Trump, who is on day two of his three-day state visit, said Mr Corbyn was a “negative force”.
“I really don’t like critics as much as I like and respect people who get things done – so I decided not to meet,” he added.
As the prime minister and US president 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 would be included in post-Brexit trade talks, Mr Trump said “everything is on the table”.
It comes after the US ambassador to the UK, Woody Johnson, last week told BBC One’s Andrew Marr Show, that he expected the “entire economy” – including the health service – to be part of a future deal between the two countries.
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”.
Health Secretary Matt Hancock was among several Conservative leadership candidates to replace Theresa May who said they would not allow the NHS to become part of any trade talks. “Not on my watch,” he tweeted.
During his visit, 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.
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.
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.
Mr Khan defended his party leader Mr Corbyn’s decision to boycott the state dinner at the palace.
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.”
Demonstrations were also planned in Birmingham, Stoke, Sheffield, Glasgow, Edinburgh, Chester, Leicester, Oxford and Exeter.
At the protest
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.
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