Kahn argues that the former Liverpool manager's identity as a builder and developer of teams is fundamentally incompatible with a dressing room dominated by individual superstars who resist tactical imposition.
The former Bayern Munich and Germany goalkeeper was speaking on Sky Germany as speculation continues to mount over who will succeed Álvaro Arbeloa at the Bernabéu.
Real Madrid are heading towards a second consecutive trophyless season, having been eliminated from the Champions League by Bayern Munich and effectively surrendered La Liga to Barcelona, who lead by eleven points with five games remaining. José Mourinho is considered the most likely appointment, but Klopp's name has featured prominently in debate across Europe.
Kahn's verdict on that possibility was blunt.
"I don't know if I would do that if I were Jürgen Klopp. Do Real Madrid and Jürgen Klopp fit together? If we analyse it more deeply, Jürgen is one of those who likes to develop, to build."
He described Real Madrid's environment as fundamentally hostile to the kind of structured, system-based football Klopp has built his reputation on.
"He works with a defined system. Real Madrid is something completely different. Real Madrid is a group of individuals, a group of superstars who flatly refuse to accept any concept being imposed on them."
Kahn pointed to the failed attempt to install Xabi Alonso as head coach — a pursuit that ended when the Basque coach opted to join Bayern instead — as evidence of how difficult the environment is to navigate even for figures of great personal authority.
"We saw it with Xabi Alonso. It is about whether, as a manager, you have the respect, can connect with the players, can manage the big stars and can convince them of the system you intend to implement."
AS reported this week that Klopp is not in fact a genuine candidate for the Real Madrid role, with the German understood to be focused on the possibility of succeeding Julian Nagelsmann with the Germany national team after the World Cup. Mourinho, meanwhile, denied on Friday that any talks with Real Madrid have taken place — a denial that, in context, has convinced few in Spain.
Real Madrid face Espanyol on Sunday evening in a fixture that feels increasingly like an audition for nobody, given the certainty that Arbeloa's tenure will end in the summer regardless of results.