Rumours spoke about Real because that's one of the only club who have the money to buy him and also because Mbappé is fan of Real.
I doubt he'll move this summer to be honest. He has clear idea in mind of best choise to do. Monaco doesn't need money and they could sell him later. Next year, the WC will take place. If Mbappé wants to win a starter place in the NT he has to make another season here at Monaco instead of staying on the bench at Madrid. Monaco fighting for the title in Ligue 1 and Ligue 1 level is rising. Next season Marseille and Lille will also have two good teams with the money they gonna spent this summer (as they have new owner). Monaco also always reach at least the quarter finals of the UCL, why moving at Liverpool ? That's sound crazy. Don't trust it.
I'd say that he not gonna move except if Madrid make a big bid (120M€ something like that) and if Mbappé ask the club to accept the bid because he wants to join them. I'm almost pretty sure he will join Madrid, soon or later, but will not leave Monaco for something else than Madrid.