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It seems that he is trying to explore his options or increase his wage demand. It's not many club to have willingness paying his high wage. Without competition to have his signature, Mbappe won't get the maximum wage demand. This is mind game in the transfer market, the big interest Real Madrid and the big desire of Mbappe to play for Real Madrid have made other clubs don't put big effort to sign him.
It's not a secret that competing with Real Madrid to sign a top player sounds almost impossible to succeed. Real Madrid's pull and financial power are unmatched by any the club right now. This fact makes the agent taking a hard time to increase their bargaining demand. Mabppe have said good bye to PSG, but other than Real Madrid, we dont see any other club putting a big interest to sign Mbappe.
That is why he was most likely playing around with two clubs: Real Madrid and PSG. The one club that wants him so badly, the other club that wants him to stay. Both are very rich although Real Madrid would not be able to pay that high of a salary, but he was smart. He made sure he can choose when to leave for no transfer fee. That means he will get the transfer fee that Real Madrid would have paid otherwise right into his pocket. Maybe he will give Real Madrid a small discount, but probably not. I guess he gets a signing bonus of around 150 million. But that's what the prices look like these days.
Given that he is among the best three players in the world, it's what he can ask for. In addition he is only 25 years old. He is probably going to be the richest footballer in the world by the end of his career.
On the one hand, Mbappe is still to play with two very rich clubs at the moment, namely PSG and Real Madrid. Mbappe places himself one level above these two clubs, at least in terms of bargaining value, as if it could be said that PSG and Real Madrid follow the rhythm of Mbappe dance. Of course, in my opinion, it is very natural, with the scale of his potential, Mbappe has the right to demand a large salary, but if Mbappe acts a lot, it is possible that Real Madrid will respond to this treatment by canceling their intention to recruit him.
In fact, Real Madrid this season has proven that without Mbappe presence they can win the La Liga trophy, and have now managed to reach the Champions League final, which I think they have the potential to win that trophy too. From another perspective, it is actually Mbappe who needs Real Madrid, in order to be able to feel the Champions League trophy and make him one of the best players in the world. So if Mbappe still demands a lot, Real Madrid certainly won't accept it. Florentino Perez is quite an expert when it comes to player transfers.