In this paper we study the mathematical model which describes quasistatic frictional contact problems between a deformable body and a rigid foundation. In this model the contact is bilateral and the behaviour of the material is described by a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a hemivariational inequality of elliptic type. Based on some recent results for the elliptic hemivariational inequality and Banach’s fixed point theorem we prove existence and uniqueness of the solution for the obtained hemivariational inequality.