The solid-liquid transitions in argon clusters of medium size (N > 55) are studied by molecular dynamics simulations on a cluster of 79 atoms. Over an intermediate range of total energy, the cluster exhibits a "dynamical coexistence" of solid and liquid states, in which thc cluster fluctuates between solid and liquid states. We propose that for medium-sized clusters, the existence of two types of structures, low-energy solid and high-energy liquid structures, leads to the dynamical coexistence. We predict that the-dynamical coexistence is a finite-size effect of a bulk melting transition.