Mr Trump was also scheduled to hold talks with Mrs May in No 10, with February 26 and 27 marked in the diary.
Downing Street refused to confirm that the visit had been cancelled on Thursday night, with a spokesman repeating the government's longstanding position that "an invitation has been extended and accepted, but no date has been set".
In 2015 Lydia Muniz, the director of the bureau of overseas building operations at the State Department, said that upgrading the existing embassy would have cost $550 million and still would not be as secure as the new embassy according to the Associated Press.
The opening ceremony may instead be hosted by US Secretary of State Rex Tillerson.
Last month, doubt was cast over whether he would come at all after the prime minister criticised him for retweeting three videos from far-right group Britain First.
The US President was due to make his first trip to Britain since being elected past year, however he took to social media to say the date was off.
Previous reports suggest that the construction of the new embassy was completely funded "by the sale of other properties in London".
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May only confirmed that Trump would be visiting on the Andrew Marr Show on Sunday but did not confirm when it would take place.
Trump's move was reported late Thursday by British media.
"You are making America great again" President Moon Jae-in told Mr Trump - echoing his famous campaign slogan.
The Prime Minister and the President clashed in November a year ago when she criticised his retweets of Britain First, saying: "I am very clear that retweeting from Britain First was the wrong thing to do".
Any visit by Trump to the United Kingdom seems certain to be met with huge protests, with 55% of Brits in one poll calling for his planned state visit to be cancelled.
Theresa May insisted last week that Mr Trump "will be coming to this country".