The 22-year-old could be available for a cut-price fee in the near future with his bumper contract set to run out in June 2022.
Klopp is close to Mbappe's father, Wilfried, and is in regular contact with the youth coach.
And L'Equipe says there's also been a little help from the club's main American backer, John W Henry.
Liverpool's owner spent some time with Mbappe's family aboard his own private jet back in 2017.
He is said to have flown them over the city of Nice for two hours while engaging in conversation with his guests.
The trip reportedly left a fantastic impression on the player's family, which could be important when Mbappe's contract is up - or sooner.