The hydrodynamic interaction between two colloids mediated by non-adsorbing polymer chains is considered for spherical colloids moving along the symmetric line. We present a resistance analysis and determine the mobility function that incorporates the polymer depletion effect. The hydrodynamic friction is calculated by solving the Stokes equation with non-uniform viscosity using the vorticity-stream function method and a bispherical coordinate transformation. The numerical results show the effective viscosity asymptotically approaches the single sphere limit of Fan et al. (T.-H. Fan, B. Xie and R. Tuinier, Phys. Rev. E., 2007, 76, 051405) for a large interparticle distance. As the particles get closer the effective viscosity decreases and finally approaches the lubrication limit, where the friction equals that of two close-approached spheres in a pure solvent. The flow analysis shows that the circulation pattern, a characteristic for the presence of the depletion layer, expands upon approach of the particles. The pressure contribution to the total drag force is determined by a competition between the lubrication force and the apparent slip effect caused by overlap of the depletion layers. The friction of the single sphere limit is only attained in the limit of an extremely small colloid volume fractions. Our results help rationalize the long-time relative diffusivity of Brownian particles in polymer solutions and this is important for a better understanding of the depletion-induced demixing kinetics.