I suspect that Nagelsmann will not be out of job for a long time considering his performance in the two last two clubs he handled. Tottenham and Real Madrid are both good clubs that he can handle and have the right players and financial support. But in terms of history and respect Real Madrid will be the best option. Carlo Ancelotti is 63years old and I think it will be more healthier for him to switch to national team coaching because club management is more tasking and stressful. Ancelotti handling the Seleção will bring a lot of improvement to the Brazilian team.
The two clubs have different histories, but Real Madrid is much better than Tottenham and Nagelsmann will be Ancelotti's successor if the management trusts him for next season. So far there is no certainty regarding the future of Nagelsmann, but Tottenham has the potential to be quicker than Real Madrid because Conte has been sacked after criticizing Spurs' management.
Thus Tottenham have the opportunity to approach Nagelsmann, although they will be competing with several clubs who have expressed interest in Nagelsmann.
I'm surprised Real Madrid is trying to negotiate with Nagelsmann. If we compare his achievement with Ancelotti, Ancelotti is still much better than him. The current bad results got by Real Madrid aren't totally caused by Ancelotti. I think the main problem is the quality of Real Madrid attackers, they are lack of quality strikers, they only rely on Benzema. To deal with the problem is signing top strikers, not changing the coach.
Sure, Nagelsmann will choose Real Madrid if he gets the offer from Real Madrid. But I assume it is just a rumor, I think Real Madrid will try to keep Ancelotti for the next few seasons.
I don't think it was because of Ancelotti's declining quality that made Real Madrid's management approach Nagelsmann, but they predicted that Ancelotti would take the opportunity to coach the Brazilian National Team.
Ancelotti has directly denied these rumors in various media, that he will stay at Real Madrid as long as management still trusts him to handle Real Madrid, so that Tottenham's chances of bringing in a coach with a German passport are much greater. PSG, Juventus and Chelsea have also expressed interest in Nagelsmann, but so far only rumors are circulating and nothing has been confirmed, it is quite interesting to see which club he will choose next season.