Trump, who arrives in the UK this afternoon for a two-day working visit - his first since his election in 2016 - will also spend time with the Queen.
Extra security has been put in place as large-scale protests are expected.
His visit comes as May publishes a White Paper setting out a blueprint for the UK's relations with the EU. Earlier this week Trump said it was "up to the people" whether the PM stayed on after two cabinet ministers resigned within hours over her Brexit policy. Trump said he had "always liked" Boris Johnson, who quit as foreign secretary.
BBC diplomatic correspondent James Robbins described the trip as "the most controversial visit ever made by an American president to Britain". Robbins said it was also "really noticeable that he will barely spend any time in London" - perhaps to avoid the protests.
May said the visit will be an opportunity to boost trade links and strengthen co-operation on security, but she also warned Trump not to ignore the "malign behaviour" of Russia when he meets Vladimir Putin in Helsinki next week.
Trump had joked earlier this week that his meeting with the Russian leader "may be the easiest" part of his European trip. Along with trade and security links, Downing Street said the other key areas to be discussed between the two leaders included Brexit and the Middle East.
Ahead of Trump's visit, May said: "As two nations - we are safer, more prosperous and more creative when we work together and I am looking forward to this week's important discussions." She said that when the UK leaves the European Union "there will be no alliance more important in the years ahead"
Downing Street insisted May welcomed Trump's decision to "engage" with the Russian leader in the Finnish capital. However No 10 also made clear that she expected him to raise issues like the Salisbury nerve agent attack on former spy Sergei Skripal and his daughter, Yulia.
May said: "Our long term objective remains a constructive relationship with Moscow. If we are to be successful in this we must keep engaging individually and as an alliance.
"That is why I welcome President Trump's forthcoming meeting with President Putin - open channels of communication between the US and Russia are key to managing the risks of confrontation."
Trump will arrive in the UK with his wife Melania on Thursday 12 July afternoon, following the Nato summit, the BBC reports.
