According to the Mirror, an overseas trip attended by May would cost tens of thousands of pounds and involve the use of the Royal Airforce luxury Voyager Jet that is reserved for such occasions.
The flippant comment has seen Trump slammed, and the sometimes vicious British press has reacted accordingly.
The Mirror labelled the comment "bizarrely un-Presidential", and pointed out the invitation sounded "more like a holiday than a state visit".
According to the Guardian, the invitation left civil servants "befuddled".
Social media users were also amused by the conversation, with one remarking "Is he offering a state dinner of a discount at the Trump hotel?"
According to the transcript, which the Prime Minister's Office has not verified, Trump also asked May to pass on his regards to the Queen during her next audience at Buckingham Palace.
He reportedly said his late mother, Mary, who was born in Scotland, was a big fan of the monarch.
During the Thursday phone call, the President-elect called the UK a "very special place for me and my country".
Trump has been doing the rounds of world leaders receiving congratulatory phone calls of since his election last week.
While those like May have waited their turn to be connected through official diplomatic channels, it was revealed yesterday that Australian Prime Minister Malcolm Turnbull jumped the queue to become the second leader to speak with Trump after his election by calling in a favour.
Turnbull had instructed US ambassador Joe Hockey to ask golfing legend and friend of Trump, Greg Norman, for the President-elect's private line.
The favour facilitated what Turnbull has repeatedly described as a "very warm" discussion.
