I don’t think Mbappe will be going to Real Madrid or Liverpool or any other team this January, to be honest. Because in these big transfers, we always hear speculations, rumors, and all that. But this time we haven't even heard anything. And another thing is big transfers like this do not happen in January transfer windows. I am not saying it’s not possible, it is possible but the chances are too low.
The question is will Real Madrid get Mbappe or any other season because we all know how he kind of betrayed Real Madrid a few years back? It all depends on Perez and his team. On the other hand, Liverpool are known to be stingy. So I don't think they will be spending this amount of money on one player. Not many clubs can afford him either, so let's see what happens.
Of course, there will be always rumors of big players, we can't avoid it. However, there is no guarantee whether the players to stay or to move. I just think that it is not impossible to leave PSG in the January transfer window. PSG can get some money and Mbappe can have a new club faster if he leaves PSG in January. But if Mbappe leaves PSG in the end of the season, PSG will get no money because Mbappe has no contract any more with SPG, he will be a free agent.
Sure, it will always depend on the decision of Perez as the president of Real Madrid club. Perez has said that Real Madrid will wait for the end of Mbappe's contract with PSG. However, Real Madrid is in urgent for a top striker. So, it is not impossible if Perez decides to sign Mbappe faster, they won't have a problem to spend some money.