A thorough study of the energy dependence of the nuclear optical potential in reactions involving the weakly bound projectiles B 8 , Be 7 and Li 6 on Ni 58 is carried out by performing a χ 2 -analysis of recent measurements of elastic scattering cross sections for energies around the Coulomb barrier. For this purpose two different potential types are used: the double folding São Paulo potential and the Woods–Saxon potential. The calculations performed with these potentials show that the energy dependences of the real V and imaginary W parts of the polarization potential satisfy the dispersion relation. Besides, within the large uncertainties, the energy dependent function W ( E ) does not seem to decrease as the energy is lowered towards the barrier as expected for the threshold anomaly usually present in tightly bound systems. Actually, the imaginary W ( E ) seems to slightly increase for energies below the barrier. On the other hand, the real polarization potential V exhibits a repulsive character around the barrier. This behaviour corresponds to the so-called Breakup Threshold Anomaly, present in the scattering of several systems with small breakup threshold projectiles as the ones of the present work. However, a definitive conclusion about the presence of the BTA cannot be made in some cases of the present analysis due to the large uncertainties of the experimental data used.