Firstly, I heard two versions of how things are now in the negotiations between Real and Mbappe:
1. Real Madrid does not make any offers but is waiting for an initiative from Mbappe, since the club’s interest is obvious, so it is waiting for a serious deal and not using itself in the auction (as was the case a year ago).
2. Real Madrid have made an offer and are waiting for a final decision from Mbappe until January 15th (apparently in the form of signing a preliminary contract, the execution of which will be very expensive to evade). If Mbappe does not agree, then Real Madrid will prepare to buy Haaland.
As for the fact that Real can easily wait until the summer when Mbappe becomes a free agent and only then sign him, I think this is not the case. One way or another, Mbappe is one of the best players of the generation and it may turn out that another strong club will get involved in the fight - maybe Liverpool, maybe Bayern (which, using the example of Kane, realized that making expensive purchases is not so bad and may decide on another top transfer). This situation will make it difficult to sign due to competition and the fact that the price for Mbappe will rise again.
The first version emerged after many rumors about Real Madrid secretly contacting Mbappe to make an offer and after that the news emerged not long from today and it makes perfect sense if Madrid didn't do that because Madrid is one of the big clubs that doesn't have to beg players.
For the second version, this rumor has just emerged and for me it also makes sense that if they send an initial contract offer before transfer window opens, maybe Real madrid will try to make an offer first to this player before it is offered by other clubs.
Yes I can understand in this situation it all depends on how Mbappe will determine the future of his career because the market price will definitely increase and this could make mbappe confused about which club to accept the offer but back to Mbappe initial plan if he not only thinks about money but also his career.
So if he really wants to have a career and get a fairly large salary, he should be able to join Madrid, accept the offer to have a career in the Champions league and I know that Mbappe really wants to get experience of lifting the Champions League trophy and this is Mbappe last chance to get a chance to be there at the right club and if this fails Mbappe prefers Liverpool or another club, Real madrid will not make another offer.