The Prime Minister will meet the US President on the sidelines of the Hamburg conference.
In talks with Mr Trump, and Japanese Prime Minister Shinzo Abe, Mrs May will make her pitch for trade deals with both countries, as well as reiterating Britain’s commitment to tackling growing aggression from North Korea.
She will also raise the issue of climate change with Mr Trump, following his decision to take the US out of the historic Paris accord.
Mrs May said she hoped Mr Trump would reverse his decision.
“I will be making that message,” she told ITV.
“I will continue to give the message that the United Kingdom believes in the importance of the Paris Agreement.”
The Prime Minister will meet Mr Abe this afternoon, where issues on the agenda will include counter-terrorism and cyber-security when the country hosts the Rugby World Cup in 2019, followed by the Olympics.
She will meet Turkish President Recep Tayyip Erdogan to talk about how to deal with the increasing number of Islamic State fighters who are using the Turkish border to move between Syria and Europe.
The biggest meeting during the first day of the summit was that between the US President and Vladimir Putin, his Russian counterpart.
Moscow described the talks, which ran over by about an hour-and-a-half, as “very constructive”.
But France’s President Emmanuel Macron earlier made it clear that he was not such a fan of Mr Trump’s leadership, using a private meeting between all the leaders to hit out at his “America first” trade policy.
Meanwhile, outside the grand meeting hall, heavily armed police are looking at another day of dealing with the thousands of protesters who have gathered outside the venue.
On Friday they used water cannon in an effort to disperse the crowd.