Spanish news outlet Fichajes claim the Red Devils could line up the 23-year-old with Real Madrid star Raphael Varane in defence.
Jose Mourinho is widely expected to take the reins at Old Trafford in the next few days following Louis van Gaal's sacking.
And it's thought that the charismatic boss will spend big as he looks to overhaul the Red Devils' under-performing squad.
Rodriguez is regarded as one of the Bundesliga's most influential defenders.
The Swiss international is a renowned set-piece taker and is the German outfit's first-choice penalty taker.
Starsport exclusively revealed earlier this month that Varane would only move to Old Trafford if Mourinho took the reins.
And with his appointment looking nailed on, £30m-rated Varane could be packing his bags.
The 23-year-old as fallen down the pecking order at the Galacticos and has struggled to hold down a starting place since Zinedine Zidane took charge.
But Varane is close to Mourinho after the Portuguese snapped him up him from Lens when he was in charge of the Galacaticos in 2011.
