If confirmed, the call could be longer than Mr Trump’s phone chat last week with Prime Minister Theresa May, which Downing Street said lasted between ten and 15 minutes.
Ukip leader Nigel Farage is understood to have spent around an hour with the president-elect at his New York home, in a visit which ruffled feathers in London.
Mrs May was the tenth world leader to receive a call from Mr Trump following his shock election on 8 November, and press reports have suggested his tone may have been more casual than she expected.
The Times reported that Mr Trump asked the Prime Minister to pass on his best wishes to the Queen and ended the conversation by saying: “If you travel to the US, you should let me know.”
Announcing details of Mrs May’s conversation earlier this week, Downing Street said that the president-elect had invited her to visit him at the earliest possible opportunity.
Asked whether the offer came in the form suggested by The Times, a Number 10 spokesman said: “The invitation from the president-elect was a very warm invitation to come and see him as soon as is possible.”
Morgan announced his chat in a message on Twitter: “Just had a 15-minute conversation with President-elect Donald Trump. He was on great form and very excited about the challenges ahead.”
The presenter of ITV1’s Good Morning Britain is a familiar face in the US thanks to his stints on TV shows America’s Got Talent and Piers Morgan Live, and also secured an interview with Mr Trump during the election campaign.
He used a column in MailOnline this week to send a message to young US voters to “cheer up” and stop complaining about the election result. He told them: “Want to know why Trump is going to be your next president? It’s because he is what’s called a ‘winner’.”