Trump was hosted by the late Queen Elizabeth II during his previous three-day state visit in 2019, which took place during his first term in office.
Traditionally, second-term US presidents are not offered a state visit and have instead been invited for tea or lunch with the monarch at Windsor Castle.
King Charles’ letter proposed a meeting to discuss details of the state visit at either Dumfries House or Balmoral, both in Scotland, a country to which Trump has connections.
BBC News understands that any visit from Trump would have to be hosted at Windsor as several Buckingham Palace state rooms are out of action due to reservicing.
Trump’s mother was born and brought up on the Hebridean island of Lewis. This year he is planning to open a new golf course in Aberdeenshire, to be named after his mother.
Speaking on Thursday, Trump said: “They’re going to do a second, as you know, a second fest… that’s what it is: a fest, and it’s beautiful, and it’s the first time it’s ever happened to one person.
“And the reason is we have two separate terms, and it’s an honour… I’m a friend of Charles, I have great respect for King Charles and the family, William, we have really just a great respect for the family.
“And I think they’re setting a date for September.”