Illgner doubts whether Klopp's personality would suit Real Madrid.
"Klopp has coached modest teams, from hard-working regions and he fits in perfectly there. The same as at Liverpool," Illgner told Radio Marca.
"Whether he would fit in at Real Madrid remains to be seen, because of the way he is.
"But yes he is a coach for Real Madrid, even though it remains to be seen whether he fits in with everything that surrounds Madrid."