According to Spanish broadcaster COPE, Mourinho called Varane and said he was prepared to hand him an £8.5m-per-year contract - more than double what he currently earns at the Bernabeu.
It is claimed Varane, 23, then discussed the situation with Real Madrid boss Zinedine Zidane and decided to stay put.
Varane reportedly also told the club he would not use United's interest to bag an improved contract.
Mourinho was in charge at Real Madrid when Varane joined from Lens in 2011.
He described the France international as one of the three best young players in the world in 2014.
Since taking charge at Old Trafford earlier this summer, Mourinho has been busy revamping his defensive options.
Ivory Coast international Eric Bailly joined from Villarreal for £30m in June while Southampton's Jose Fonte remains a target.
Stoke target Phil Jones has been told he has a future at Old Trafford despite being excluded from Mourinho's first squads while Marcos Rojo is available for sale.
