The 17-year-old was thought to be on the verge of joining the Spanish giants on a five-year deal, but the Sweden international has opted for Thomas Tuchel's side at the Westfalenstadion.
The starlet, who is said to have cost Dortmund £8m, arrives in the Bundesliga highly rated having scored 10 goals in 24 appearances in Sweden's Allsvenskan league.
"Alexander Isak is a high-calibre young talent who was wanted by numerous top European clubs," The Mirror quotes Dortmund's general manager Michael Zorc as saying. "We are very glad that he has decided to join Borussia Dortmund."
Isak, who became Sweden's youngest goalscorer ever after making his senior international debut last week, broke through AIK's academy just last year.