We study the one-loop gluon polarization tensor at zero and finite temperature in the presence of a magnetic field, to extract the thermomagnetic evolution of the strong coupling alpha(s). We analyze four distinct regimes, to wit, the small and large field cases, both at zero and at high temperature. From a renormalization group analysis, we show that at zero temperature, either for small or large magnetic fields, and for a fixed transferred momentum Q(2), alpha(s) grows with the field strength with respect to its vacuum value. However, at high temperature and also for a fixed value of Q(2), we find two different cases. When the magnetic field is even larger than the squared temperature, alpha(s) also grows with the field strength. On the contrary, when the squared temperature is larger than the magnetic field, a turnover behavior occurs, and as decreases with the field strength. This thermomagnetic behavior of alpha(s) can help explain the inverse magnetic catalysis phenomenon found by lattice QCD calculations.