Mbappe, with a deal to 2022, has informed PSG management that he intends to make this season his last in Paris.
Real Madrid are regarded as favourites for his signing, though AS says the Reds manager still 'dreams' of bringing the striker to Anfield.
And Klopp is working hard to convince the youngster about a move to Liverpool.
The German enjoys a good relationship with Mbappe and particularly his father, Wilfried, and maintains regular contact with the family.
Klopp tried to sign Mbappé both in 2016 and in 2017 - and has convinced his board to try again when the youngster becomes available next year.