C60 polymerized carbon nanotori and haeckelites nanostructures are investigated when they are exposed to an external magnetic field (B). A π-electron tight binding model in conjunction with the London approximation has been implemented in order to calculate the ring currents produced by the magnetic field B and thus, we determined the diamagnetic or paramagnetic response as a function of the carbon atomic arrangement. We have found that magnetic response depends strongly of the morphology of the carbon nanostructures. The results obtained for corrugated rings formed by C 60 joined by 2-fold, 3-fold, and 5-fold symmetry axis. In this case, we found that when C60 is joined, by the 5-fold axis (by means of pentagonal rings), nanotori structures exhibit strong current fluxes that favor the generation of large magnetic moments perpendicular to the ring plane. When the C60 is joined trough the 2-fold or 3-fold axis of symmetry, the corrugated rings exhibit interesting ring patterns which depend of the number of C60 molecules. In addition, electronic and magnetic properties of haeckelite type tori are discussed.