The Sunday Star reported that Jose Mourinho will settle for Draxler after Dortmund refused to sell any more of their star players following the departure of Ilkay Gundogan to Manchester City.
The new United boss is looking to revitalise his attacking options, although Wolfsburg are reluctant to sell the 22-year old Draxler given that he only arrived in Lower Saxony last year.
The Red Devils' interest comes despite Wolfsburg’s insistence they will not listen to offers for Draxler and Andre Schurrle.
Wolfsburg sporting director Klaus Allofs said earlier this month: "We always say that we listen to everything. That expressly does not apply for Andre and Julian.
"We don't want to sell them, we want to return to the Champions League with them."
Left wing Draxler began his career at Schalke where he made 118 appearances before switching to Wolfsburg at the beginning of last season.
Draxler is currently in France with the Germany squad at Euro 2016.