We compute the one loop right and left anomalous tensor couplings (gR and gL, respectively) for the top quark, in the aligned two-Higgs-doublet model. They are the magnetic-like couplings in the most general parameterization of the tbW vertex. We find that the aligned two-Higgs doublet model, that includes as particular cases some of the most studied extensions of the Higgs sector, as Type I and II 2HDM, introduces new electroweak contributions and provides theoretical predictions that are very sensitive to both new scalar masses and the neutral scalar mixing angle. The model can account for new CP violation effects via the introduction of complex alignment parameters that have important consequences on the values for the imaginary parts of the couplings. The top anomalous tensor couplings will be measured at the LHC and at future colliders providing a complementary insight on new physics, independent from the bounds in top decays coming from B physics and b→sγ.