Trump's first official visit to Britain a year ago was overshadowed by criticism of May's approach to Brexit, as well as large demonstrations.
"As the Prime Minister has said previously, we were disappointed by the USA decision to withdraw from the Paris Agreement in 2017 and continue to hope they will return", the spokesman said.
Embattled UK prime minister has announced she will resign on June 6, after the parliament observed that her Brexit deal failed thrice to get passed.
"If they don't get what they want, I would walk away", he said. "If you don't get the deal you want, if you don't get a fair deal, then you walk away".
In particular, Trump is expected to push the government on the need to exclude Huawei from the UK's 5G network, with some Tory MPs up for the leadership having already said that they would consider scrapping their relationship with Huawei after USA ambassador Woody Johnson said it was "a big risk".
Trump will meet with the PM on Tuesday and Wednesday and he is also expected to sit down with candidate Boris Johnson and with Brexit Party leader Nigel Farage.
"I like Nigel a lot".
Speaking to The Sun, Mr Trump, 72, said: "It's something I find very interesting".
Pompeo: Humanitarian aid plan for Iran "unproblematic"
But earlier this week he made a U-turn and urged all Germans to wear skullcaps in solidarity with the Jewish community. Pompeo and Maas also tackled the thorny topic of Huawei. "None of us should shrink in the face of prejudice".
"He is a very smart person" with a "lot to offer", said Trump, although he acknowledged that British authorities "won't bring him in". "They just haven't figured that out yet".
On the divorce bill - Britain's liabilities as it leaves an economic and political bloc it has been part of since 1973 - Trump told the newspaper: "If I were them I wouldn't pay $50 billion. That is a tremendous number".
Trump also said he would have "to know" veteran Socialist Labour opposition leader Jeremy Corbyn before authorizing United States intelligence to share its most sensitive secrets with a hard-left government.
Huawei will top talks in London after the British government appeared to defy Trump administration demands and allow the Chinese company a limited role in building 5G networks.
"Guess who's been busy today mowing a stiff message for Trump under the Stansted flightpath", tweeted Nancarrow, a student who is also involved with an online eco-marketplace.
Britain's so-called special relationship with the United States is an enduring alliance, but some British voters see Trump as crude, volatile and opposed to their values on issues ranging from global warming to his treatment of women.
The three-day trip was cast as a chance to celebrate Britain's "special relationship" with the United States, boost trade links in a post-Brexit world and reaffirm security cooperation ahead of the 75th anniversary of the allied D-Day landings in World War Two.
Hundreds of thousands protested against him during a trip past year and a blimp depicting Trump as a snarling, nappy-clad baby will fly outside Britain's parliament during the visit.
Labour leader Jeremy Corbyn, a veteran socialist, denounced the president's "entirely unacceptable interference in our democracy".