In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $\mathbb{K}= \mathbb{K}^{+} \oplus \mathbb{K}^{-}$ of the Krein space $\mathbb{K}$ with $\mathbb{K}^{+}$ and $\mathbb{K}^{-}$ invariant under $T$.