Mourinho has kept his interest in Varane secret until United's bid emerged, but is now working overtime to lure him to Real Madrid.
The Special One, who clashed with Fergie numerous times when he was Chelsea boss, would love to pip the Scot again.
Mourinho has invited Varane to Madrid on Tuesday and will give him a personal tour of the Bernabeu and Real's training ground.
It is understood he is willing to outbid United and can offer bigger wages than the Premier League champions.
French legend Zinedine Zidane has also told Varane to join the Spanish giants. Zidane, who is now Real's sporting director, will be involved in the talks next week.
Fergie won't give in, but knows he could be fighting a losing battle.