Their relationship goes beyond that of a teacher and student, though it is not romantic. Kojou frequently refers to her as Natsuki-chan, much to her dislike. They care for each other in a fashion that is best described as friendship. She is also one of the few people who knows Kojou is the fourth Primogenitor.
Natsuki appears to know Koyomi's true identity as one of the leaders of the Lion King Organization.