The Executive Office has confirmed an invitation was sent to the American leader.
It was sent the day after the tycoon's victory in the election.
The paper's report, by Rodney Edwards, says the letter was sent on November 9, 2016, the day after the presidential election.
“We are writing to offer our personal congratulations and those of all the people we represent on your election as the 45th President of the United States of America," the letter to the President-elect says.
"We are a small region but we are fortunate to have strong historical, economic and political ties to the United States. Some of those links predate the War of Independence and we can boast that no fewer than seventeen of your predecessors had Scots-Irish heritage."
The letter went on to say how proud the leaders were to have the US as the largest inward investor to Northern Ireland, adding: "We wish you every success in your new role and we extend an invitation to visit Northern Ireland. You can be assured of a warm welcome.”
Signed by both Mrs Foster and Martin McGuinness, they end: "Best wishes".
A planned State Visit to the UK has sparked controversy. More than one million people have signed a petition calling for the invitation from the Queen to be withdrawn.
However, a counter petition backing the visit has been signed by more than 100,000 people.