We compute the one loop right and left anomalous tensor couplings (g(R) and g(L), 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, introduces new electroweak contribution's and provides theoretical predictions that are very sensitive to both new scalar masses and the neutral scalar mixing angle. For a largo area in the parameters space we obtain significant deviations in both the real and the imaginary parts of the couplings gR and gL, compared to the predictions given by the electroweak sector of the Standard Model. The most important ones are those involving the imaginary part of the left coupling g(L) and the real part of the right coupling gR. The real part of g(L), and the imaginary part of gR also show an important sensitivity to new physics scenarios. The model can also 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 -> s gamma.