The 18-year-old enjoyed a breakout season in 2016-17, with a 26-goal haul seeing him earn admiring glances from across Europe and a £100 million price tag.
That valuation has done little to curb Madrid’s interest, as the La Liga champions are always in the market for global stars with the potential to shine for them on and off the field.
Dugarry believes the Spanish giants need to do all they can to get a deal done as they could live to regret it if Mbappe ends up joining a European rival.
The former France international told RMC on the future of his highly-rated countryman: “There is a real decision to be made.
“But sincerely if Mbappe doesn’t go to Real Madrid, it will be a mistake by Real Madrid."
He added: “He is the new gem of world football and with Ronaldo’s age, he must end up at Real Madrid," he added.
“They are the best club in the world.”
Mbappe is under contract at Monaco until 2019 and recently revealed that he will give no thought to his future until he returns from a well-earned summer break.
He said: "We're going to go away with the family and we'll think about what's going to happen.
“We have to sort it out, see what happens with the club too, because I'm under contract with a club, I'm not free. We'll see what's happening.”