Dortmund released a statement saying they have come to an agreement "in principle" with the 24-year-old and his current club Shakhtar Donetsk, although "the details need to be sorted out".
The news will come as a blow to Liverpool, who were also interested in Mkhitaryan who scored 25 goals in 29 Ukrainian Premier League matches last season.
He is expected to cost Dortmund €23m, according to press reports, and will sign a four-year deal.
Dortmund finished second in the Bundesliga last season and can look forward to another campaign in the UEFA Champions League, where they reached the final in May before losing to Bayern Munich.
Borussia see Mkhitaryan as the ideal playmaking replacement for Mario Gotze, who left to join Bayern at the end of last season.
Mkhitaryan's signing could spell an end to Dortmund's transfer dealings during the off season as the statement said the stock-exchange listed club had "exhausted" their finances.
Last week, Dortmund completed the signings of Gabon forward Pierre-Emerick Aubameyang from Saint-Etienne and Greek defender Sokratis Papastathopoulos from Werder Bremen.