Recent measurements for fusion cross section at energies around the Coulomb barrier have systematically indicated a significant deviation of fusion cross sections from a prediction of double-folding model. It has been argued that the deviation can be accounted for if one uses a large value of surface diffuseness parameter for a Woods-Saxon internuclear potential. We investigate this problem using large-angle quasi-elastic scattering, that is a good counterpart of fusion reaction. Towards a reconciliation of the apparent anomaly in the diffuseness parameter for fusion reactions, we also discuss possible ingredients which are missing in present nuclear reaction models. Those include a deviation from the frozen density approximation, the effect of antisymmetrization and the role of coordinate dependent moment of inertia for the relative motion between the colliding nuclei.