For a proper smooth variety over a perfect field of characteristic p, crystalline cohomology is a good integral model for rigid cohomology and crystalline Chern classes are integral classes which are rationally compatible with the rigid ones. The overconvergent de Rham-Witt complex introduced by Davis, Langer and Zink provides an integral p-adic cohomology theory for smooth varieties designed to be compatible with rigid cohomology in the quasi-projective case. The goal of this talk is to describe the construction of integral Chern classes for smooth varieties rationally compatible with rigid Chern classes using the overconvergent complex.