The Sunday Times reports the Queen, 90, plans to invite Mr Trump and his wife Melania to Buckingham Palace after his inauguration in January.
Discussions are reportedly due to begin soon to organise a date for June or July next year.
The Telegraph reports the US President-elect talked to Prime Minister Theresa May on the phone last week, telling her his late Scottish mother had been a "big fan" of the Queen.
Meanwhile, new figures show Mr Trump is the 'biggest loser' of any US President-elect in history.
Hilary Clinton's popular vote margin is the biggest for any losing candidate. She is now ahead of him by more than 1.5 million votes and counting.
The Associated Press count as of Saturday had Ms Clinton on 63,390,669 and Mr Trump behind on 61,820,845 - a difference of 1,569,824.
It's not the first time the Electoral College system has turned the tables on the leading Presidential hopeful.
Democratic candidate Al Gore received 543,816 votes more than George W Bush in 2000. The Republican candidate won the election however, after a controversial victory in Florida gave him the 270 Electoral College votes required.
Mr Trump criticised the Electoral College system in 2012, tweeting: "The electoral college is a disaster for a democracy."