The £30m buy from Villarreal made his first Manchester United appearance in Saturday’s 2-0 friendly win at Wigan.
Centre back Bailly, 22, barely broke sweat in his 82 minutes on the pitch.
But new United boss Mourinho saw enough to be impressed and has vowed to make the Ivory Coast international the complete defender.
“Of course he is a very young boy coming from a completely different environment, but you could see what he is at this moment,” said Mourinho who took France international Varane from Lens to Madrid as an 18-year-old in 2011 and has also worked with defensive greats like Real’s Sergio Ramos and Chelsea captain John Terry.
“I will give him everything to try to make him better, like I did with other central defenders I’ve had in my hands like, for example, Varane.