The Daily Mail says United manager Sir Alex Ferguson has sent scouts to watch the attacking midfielder four times and that two scouts were despatched to Germany's recent game against Holland.
Arsenal feared they would be quoted at least £30million, after admitting their interest last month, and the teenager's valuation has soared even higher after enquiries from Real Madrid and some of Italy's leading clubs.
Responding to the interest from United and Arsenal, Dortmund's top administrator, managing director Hans-Joachim Watzke, told them to forget any thought of trying to sign him towards the end of the January window, when he returns to fitness, or even at the end of the season.
He even appeared to aim a warning shot in the direction of Manchester City owner Sheikh Mansour, as he said: "All I can say to all these clubs is they should save themselves the cost of a phone call, because they will be wasting their time and money contacting us. Even if someone rides out of the desert and makes us an offer, it would make no difference. Mario is staying here."