Rojo joined Man United last season from Sporting Lisbon but injuries have meant that the Argentine has failed to string together a consistent run in the side.
The 25-year-old, who watched United’s 1-0 win against Liverpool at Anfield last week among supporters in the away stand, is currently sidelined with a dislocated shoulder.
Schalke have reportedly identified Rojo as a replacement for their injured captain, Benedikt Howedes, who is facing three months on the sidelines.
The German club’s head coach, Andre Breitenreiter, is not said to be put off by Rojo’s injuries.
Louis van Gaal has repeatedly had to patch together a makeshift defence this season as the likes of Luke Shaw and Antonio Valencia are also injured. It is highly unlikely that United will let go of Rojo unless a suitable replacement is signed. When he has been on the pitch, Rojo has impressed and the hope is that he can recover fully before kicking on at Old Trafford.