The 23-year-old France International has not been able to break into Zinedine Zidane's side this season, with Sergio Ramos and Pepe preferred in the centre of the Real defence.
According to The Mirror, United are preparing a R546-million bid for the former Lens man, who is on the hunt for first team football.
Former Red Devils boss Sir Alex Ferguson had previously tracked Verane in 2011, but instead sogned Phil Jones from Blackburn.
With Louis Van Gaal's side's hopes of a UEFA Champions League spot hanging by a thread, with just one game to go, it could play a huge part in whether they convince Verane to join ahead of next season.