The German World Cup ace has been a target for the Red Devils for some time, with manager Louis van Gaal rumoured to have made him his prime target to replace departed captain Nemanja Vidic.
But despite numerous overtures towards the 25-year-old, Dortmund have held firm.
Yet United are plotting one last ‘take it or leave it’ offer in the region of £20m for the defender.
Hummels is understood to be happy in his homeland, but if he is allowed to talk to the Premier League giants he will give a switch serious consideration.
The rugged centre-back has been one of Dortmund’s stars in recent seasons, playing a big role in their most recent Bundesliga success and run to the 2013 Champions League Final.
Should United be unable to tie up a deal, Arsenal’s Thomas Vermaelen, Stefan de Vrij of Feyenoord and fellow Rotterdam star Bruno Martins Indi are also on LVG’s list as he plans to bring in two new players for the position.