Nerazzurri chief executive Beppe Marotta confirmed over the weekend that a deal was on the verge of being struck for the 21-year-old, who reportedly arrived in Milan for a medical on Wednesday (AEST).
Hakimi has been a star for Dortmund this season, having joined on a two-year loan from LaLiga giant Madrid in 2018.
And he said a fond farewell to the Bundesliga club ahead of his proposed switch to Serie A.
"The time has come to close a very beautiful stage in my life. After two wonderful years, I'm ready to leave this club that has given me so much joy," Hakimi wrote on Twitter and Instagram.
"From the first moment, the care and attention by the club and the city has been amazing. Now I know that I was right to choose Dortmund as my home two years ago.
"I want to thank all my teammates, the coaching staff, and all the club's staff for the love and trust they have shown me at all times.
"I also want to thank the wonderful fans who have supported us unconditionally in every game.
"I'll never forget the yellow wall (Glebe Wand) which is simply wonderful.
"Finally, I would like to thank my family and friends for supporting me at all times.
"I hope that one day our paths will cross again. See you soon Borussia Dortmund!"