Mr Trump told the Prime Minister she could visit him as soon as possible in Washington after their first talks in which they affirmed the “special relationship” between the two countries.
Downing Street confirmed that the two also “affirmed” the importance of the “special relationship” in a phone call.
The Prime Minister and the business tycoon have already discussed their desire to strengthen trade ties and confirmed the two countries would remain “close allies”, a Number 10 source said.
Mr Trump said it would be a “great honour” to welcome Mrs May to Washington, the source said.
A Downing Street statement said: “President-elect Trump… added that the UK is a ‘very, very special place for me and for our country’.”
The statement echoes comments made by Chancellor Phillip Hammond at a press conference in which he said the special relationship between Britain and the US was alive and well.
Donald Trump has strengthened the special relationship with his invitation to Mrs May
It is understood Mr Trump and Mrs May spoke shortly before 2pm over the phone today, with Mrs May congratulating Mr Trump on his hard fought campaign.
The pair have discussed a desire to strengthen trade and investment ties, which could be significant as the UK hopes to strike deals around the world after leaving the European Union.
Mrs May congratulated Mr Trump on his election victory yesterday despite previously criticising his remarks.
Mr Trump has invited Mrs May to Washington DC
She said: “I would like to congratulate Donald Trump on being elected the next president of the United States, following a hard-fought campaign.
“Britain and the United States have an enduring and special relationship based on the values of freedom, democracy and enterprise.
“We are, and will remain, strong and close partners on trade, security and defence.
Theresa May had previously criticised Mr Trump’s comments about Muslims
“I look forward to working with President-elect Donald Trump, building on these ties to ensure the security and prosperity of our nations in the years ahead.”
Mrs May previously slammed Mr Trump’s comments on Muslims as “divisive, unhelpful and wrong”.
Mr Trump likened his election success to Brexit – calling it ‘Brexit plus plus’.
Mrs May was the first person Mr Trump invited to meet him
Senior republican commentators have said that Mr Trump’s shock election will be highly beneficial for post-Brexit Britain.
Last night Ukip leader Nigel Farage, who joined Mr Trump on the campaign trail, said the Trump victory was great news for Britain after President Barack Obama threatened to put the UK “at the back of the queue” for daring to leave the EU.
Mr Farage said: “We now have a US president who likes our country and understands our post-Brexit values.”
Dan DiMicco, Mr Trump’s adviser, said last month Britain would be first in line for a trade deal.
Mr Trump has made clear he wishes to sustain and improve the special relationship with Britain as it exits the EU.
Charlie Wolf, a British-based Republican commentator and radio host, also believes the EU will face an uphill battle with President-elect Trump.
Hillary Clinton conceded defeat to Mr Trump in an emotional speech yesterday
Speaking to Sky News he said: “This is part of the good news [Trump’s relationship with the UK].
“People were all saying the world is going to fall apart. It is not. This is a guy who understands trade and depends on trade.”
Mr Trump’s resounding Republican victory – winning the presidency by surging past the symbolic 270 electoral college votes needed to win – was sealed yesterday morning.