We consider two mathematical models, which describe the frictional contact between a deformable body and a foundation. In both models the process is assumed to be quasistatic, the material is viscoelastic, and the friction is given by a subdifferential boundary condition. In the first model the contact is described with a univalued condition between the normal stress and the normal displacement and in the second model it is described with a subdifferential condition, which links the normal stress and the normal velocity. For each model we derive a variational formulation, which is in the form of a history-dependent hemivariational inequality for the velocity field. Then we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proof is based on a recent result on history-dependent hemivariational inequalities.