The Red Bull Salzburg attacker, 20, is set to follow Erling Haaland's path from Austria to Germany.
Adeyemi caught the eye of United after scoring 23 goals in 42 appearances this season, but the young German has decided to ply his trade at Dortmund.
The German's agent, Thomas Solomon, said: "We're now close to completing the deal with Borussia Dortmund.
"There will be no release clause included into his five-year deal.
"Rangnick? Yes, Manchester United wanted to enter the race - but we want BVB."
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