The 7Li+28Si elastic scattering was studied at near-barrier energies, namely, 8, 8.5, 9, 10, 11, 13, 15, and 16 MeV, with the aim to map the real and imaginary part of the optical potential and therefore probe the threshold anomaly. Angular distributions were measured over a wide angular range of (θlab=25° to 150°) for the lower energies and of (θlab=10° to 100°) for the higher energies. The present data, together with previous ones on heavier targets (138Ba and 208Pb) at near barrier energies, were analyzed by using optical potentials obtained in a double-folding framework. The results were compared with previous measurements of 6Li on the same targets. It was found that a striking difference occurs between the imaginary potentials of 6Li and 7Li, which, respectively, present an increasing and decreasing behavior approaching the barrier from higher to lower energies. On the other hand, this energy variation is not fully reflected to the real part of the potential, as it is described by dispersion relations. The strength of the real potential remains almost constant with a weak declining and uprising trend for the 6Li and 7Li, respectively. For a better understanding of our results, continuum-discretized-coupled-channel calculations were also performed and are discussed.