A dynamical system is saturated when for any invariant measure μ, the topological entropy of the set of the μ—generic points equals the measure-theoretic entropy of the system. This fact was confirmed by Fan, Liao and Peyrière for systems with specification. In a recent article we extended this result under the condition of non-uniform specification. In this work we consider another weaker condition than specification called almost specification property. This concept was introduced by Thompson as a modification of the almost property product by Pfister and Sullivan. We prove herein the saturatedness of systems under the Thompson condition. The saturatedness is a key point to establish a variational principle for V—statistics, as was developed by Fan, Schmeling and Wu.