The clarification comes after Trump said he had cancelled his trip to London next month to officially open the new USA embassy in protest over its new location south of the River Thames, and the sale of the old building, in the upmarket Mayfair district.
Downing Street said no date has been confirmed for any visit by Mr Trump and that the opening of the USA embassy "is a matter for the US".
In response to the cancellation, London Mayor Sadiq Khan celebrated the news, saying the USA president "is not welcome here". Mr Johnson suggested that Sadiq Khan, the mayor of London, and Jeremy Corbyn, the Labour leader, had scuppered the chance of a visit from the United States president in an effort to draw a partisan line on the issue.
May proclaimed the strength of the "most special relationship" between the two countries and the government extended an invitation for a state visit as a guest of Queen Elizabeth II.
"As a junior foreign service officer, I signed an oath to serve faithfully the president and his administration in an apolitical fashion, even when I might not agree with certain policies".
A Downing Street spokesman said Mrs May would tell Mr Trump he is welcome in London.
"Let's hope that Donald Trump also resists the pursuit of his divisive agenda".
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Some British lawmakers questioned whether Trump would be welcome in London because of previous tweets and criticism of Muslims and his sniping at London Mayor Sadiq Khan in the aftermath of a terror attack in that city past year. "It seems he's finally got that message", the London mayor tweeted. Mr Khan, a member of the main opposition Labour Party, said there would have been "mass peaceful protests", and that it was a "mistake" to invite him.
Downing Street was unable to say whether Mr Johnson was speaking for the Government, stressing how it was "not aware of the circumstances of the tweet".
Dan Mahaffee, senior vice president and director of policy at the Center for the Study of the Presidency and Congress, agreed that Trump canceled the trip likely because his arrival in London would be spoiled by protests by the British public.
The "strong and deep" partnership between Britain and the U.S. "will endure" despite Mr Trump's cancellation, a spokesman added.
But the date has yet to be set in the face of deep hostility to the president in Britain, prompting speculation it could be turned into a lower profile trip focused around the opening of the new embassy.
British Prime Minister Theresa May's Spokesperson Max Blain said Vauxhall is a vibrant part of London and "home to many businesses", and not an "off location" as Trump had described.
"We are still working with our United Kingdom allies to find a date for a visit for the invitation that was offered and accepted, " she said Friday.