This website understands the 17-year-old attacking midfielder has agreed on a lucrative three-year deal with a two-year option with the Bundesliga giants.
The Ajax development product is set to link up with Dortmund in Germany after he turns 18 in September.
"Look, he's finally leaving and I'm very happy for the player," a source close to Matthews' family told snl24.com/kickoff.
"Let's make no mistake, he's a very talented player and I hope he makes his mark.
"Obviously he lacks the physicality and he needs to bulk up more for the European game but there's no disputing his talent."
Matthews also previously had a training stint under Cape Town City coach Benni McCarthy.
The Urban Warriors could not be reached for comment.