Recent studies show growing interest in materials with asymmetric friction forces. We investigate terminal motion of a solid body with a circular contact area. Assuming that friction forces are asymmetric orthotropic, we consider Hertz and Boussinesq laws for pressure distribution. Equations for friction force and moment are formulated and solved for both cases. Numerical examples demonstrate significant impact of the asymmetry of friction on the motion. Our results can be used for more accurate prediction of contact behavior for bodies made of new materials with asymmetric surface textures.