The 21-year-old defender has previously hinted at a move to United, but reports coming in today suggest that Chelsea, Juventus, Bayern Munich, Monaco and West Ham are also keen on the Stuttgart player.
According to reports elsewhere, Juventus have been keeping a close eye on the German youngster and have scouted the player for some time with the intention of a bid to be placed in January.
Monaco failed with a £12 million offer in the summer transfer window.
Rudiger spoke out about the possibility of a move to United last month after revealing his desire to play in the Premier League.
“Of course I am aware of the interest from Manchester United,” he said.
“They are definitely one of the top clubs in Europe and they are in a league I have been dreaming of. The Premier League is very physical, intense, breathtaking and with a lot of goals.