Abstract: The laser-modulated magnetopolaron effect on shallow donors in semiconductors is investigated in the presence of a magnetic field in the Voigt configuration. A nonperturbative approach is used to describe the electron-photon interaction by including the radiation field in an exact way via a laser-dressed interaction potential. Through a variational approach we evaluate the donor binding energy. We find that the interaction strength of the laser-dressed Coulomb potential in the z direction cannot only be enhanced but also weakened by the radiation field, while that in the x-y plane is only weakened. In this way, the binding energy of the states with odd z parity, like 2p(z) can be decreased or increased with respect to its static binding energy by the radiation field, while that of the other states can be only decreased. Furthermore, all binding energies become insensitive to the magnetic field if the radiation field is strong. The magnetopolaron effect on these energies is studied within second-order time-dependent perturbation theory. In the nonresonant region, a laser-modulated magnetopolaron correction, including the effect of single-photon processes, is observed. In the resonant region, a laser-modulated magnetopolaron effect, accompanied by the emission and absorption of a single photon, is found. Moreover, the 1s -> 2p(+) transition, accompanied by the emission of a single photon, is tuned by the radiation field into resonance with the longitudinal-optical phonon branch. This is electrically analogous to the magnetopolaron effect, and therefore we name it the dynamical magnetopolaron effect. Finally, by changing the frequency of the radiation field, these interesting effects can be tuned to be far away from the reststrahlen band and, therefore, can be detected experimentally. This in turn provides a direct measure of the electron-phonon interaction.