The Brazil under-23 international moved to Madrid from Flamengo in January for €30 million.
Sources told ESPN on Saturday that the transfer — which does not include an option to buy for the Bundesliga club — was on the verge of completion.
Reinier, 18, will now join up with his new Dortmund teammates after they returned from a preseason training camp in Switzerland this week.
The playmaker is yet to feature for Madrid’s first team as the squad’s three non-EU player spots are occupied by fellow Brazilians Vinicius, Rodrygo and Eder Militao.
Reinier started three times for Madrid’s reserve team Castilla in February and March, scoring three goals, before Spain’s Segunda B season was abandoned due to the coronavirus pandemic.
At Dortmund he will complete an exciting attacking lineup for next season, playing alongside other talented youngsters such as Erling Haaland, Jadon Sancho and new boy Jude Bellingham.
The 2020-21 Bundesliga campaign is due to start on Sept. 18.
“He’s a ‘nine-and-a-half’ who can play as a false nine, or on the wing,” BVB coach Lucien Favre said on Monday. “I don’t know if he can play as a number eight in a 4-1-4-1. He’s very young.
“Our attack looks like a high school now! From what I’ve seen, he’s really good in attack and at finishing. At 18 he can still improve a lot.”