We propose a novel framework for predicting the evolution of dynamical systems by learning the Koopman operator in the space of linear functionals on the Signature transform of trajectory data. The Signature, a central object in rough path theory, provides a universal and compact representation of paths through iterated integrals, enabling linear models to approximate a wide class of non-linear functionals. By restricting observables to lie in the span of truncated Signatures, we construct a finite-dimensional approximation of the Koopman operator, which we estimate directly from data using regularized linear regression. This approach merges the expressiveness of operator-theoretic methods with the structural richness of Signature features.