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.
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."
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