Fusion excitation functions have been measured for the first time with rather good accuracy for 32S+90Zr and 32S+96Zr near and below the Coulomb barrier. The sub-barrier cross sections for 32S+96Zr are much larger than for 32S+90Zr. A coupled-channels calculation considering the inelastic excitations is capable of describing sub-barrier enhancement only for 32S+90Zr. The unexplained part for 32S+96Zr is found to be correlated with the positive-Q-value intermediate neutron transfers in this system. The comparison with 40Ca+96Zr suggests that couplings to the positive-Q-value neutron transfer channels may play a role in the sub-barrier fusion enhancement. Multi-neutron transfers are taken into account in Zagrebaev’s semiclassical model to explain the discrepancies of the sub-barrier fusion cross sections for 32S+96Zr.