Theresa May is facing an embarrassing claim that Donald Trump has not formally invited her to visit him in the White House.

Last week, Downing Street said the President-elect had urged the Prime Minister to meet him in Washington “as soon as possible” after his inauguration next year.

The invitation was hailed as proof that the so-called ‘special relationship’ is still alive – after it was revealed that he spoke with nine other world leaders before Ms May.

But Downing Street has now refused to deny that the actual words used by Mr Trump were “If you travel to the US, you should let me know.”

The extraordinarily casual phrase – revealed in a leaked transcript of last Thursday’s telephone conversation, seen by The Times – is said to have stunned civil servants.

It raised the question of whether Mr Trump had, as claimed, proposed a formal state visit by Ms May soon after he assumes power.

Overseas visits by the prime minister to the US cost hundreds of thousands of pounds and can take months to arrange.

A No.10 spokesman refused to discuss the words used by Mr Trump, saying: “We gave a readout at the time – I am not going to elaborate on that anymore.”

Asked if the invitation had been “informal and a bit embarrassing”, he added: “The invitation from the President-elect was very warm invitation to come and see him as soon as is possible.”

Last week, No.10 was left scrambling to face down criticism that the pair’s delayed conversation had left Britain somewhere near the “back of the queue”.

Before contacting Ms May, Mr Trump managed to speak with the leaders of Egypt, Ireland, Mexico, Israel, Turkey, India, Japan, Australia and South Korea.

Mr Trump also disclosed that he was a big fan of the Queen and asked for his regards to be passed on to Britain’s longest reigning monarch, The Times reported.

And he recalled the alliance between Ronald Reagan and Margaret Thatcher as a template for his relations with the Prime Minister - but told her theirs would be better.

No.10 said Ms May stressed how the close allies and would work together on security issues and how she wanted to strengthen bilateral trade and investment.