Trump claimed he had scrapped the visit because he didn't want to open the new U.S. embassy in the British capital, but Khan suggested the real reason was because he wouldn't be welcome.
Trump said he had cancelled next month's visit, which had been set to include the opening of the new USA embassy in London, because he did not like the building and thought it was a "bad deal" financially.
The decision to move the embassy from Grosvenor Square to Nine Elms was made in 2008 under George W. Bush, not Obama.
The clarification comes after Trump said he had cancelled his trip to London next month to officially open the new United States embassy in protest over its new location south of the River Thames, and the sale of the old building, in the upmarket Mayfair district.
Some U.K. lawmakers had said Trump was not welcome in Britain after he re-tweeted videos from a far-right British group and criticized London Mayor Sadiq Khan following a terror attack previous year.
Asked if it was a discourtesy to the Queen for Mr Trump to keep putting off the state visit, he replied: "An invitation has been extended and accepted and we will set out details in due course".
"The project has been executed within the established budget", an embassy spokesman said, adding it had "considered more than 50 sites" in London ahead of the move, which cost up to $1billion and "used no tax dollars".
He tweeted late on Thursday that he was not a big fan of the Obama administration that sold "perhaps the best located and finest embassy in London for "peanuts", only to build a new one in an "off" location for $1.2 billion, Xinhua news agency reported.
"I think we have to welcome the American President to Britain", Johnson said.
In January previous year, Mrs May was the first foreign leader to visit Mr Trump in the White House.
It does not mean plans for the US President's state visit to the United Kingdom are off entirely, with a Downing Street spokesman telling the newspaper: "An invitation for a state visit has been extended and accepted". The uproar came after Trump re-tweeted three anti-Muslim videos posted by a leader of the far-right group Britain First - a tiny group that regularly posts inflammatory videos.
Trump's cancellation was met with humour on social media, with parodies of the president's tweet trending on Twitter.
He noted how Mr Trump would have been met by "mass peaceful protests".
His comments sparked criticism from Foreign Secretary Boris Johnson, who accused the Mayor of London of endangering the so-called "special relationship".
"It seems he's finally got the message", wrote London's mayor, Sadiq Khan. "We will not allow US-UK relations to be endangered by some puffed up pompous popinjay in City Hall".