The German attacking midfielder moved from Bayer Leverkusen for a €25 million fee.
The Manchester Evening News says United asked Leverkusen about whether he was available before he completed the move.
However, they were told by the German club that everything with Dortmund was already agreed.
The player had committed and is happy to move to BVB for the coming seasons.
United are chasing several players as they look to improve on a sixth placed finish this season.