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Vertical-cavity surface-emitting lasers (VCSELs) with optical feedback are known to exhibit different transverse-mode regimes depending on the injection current. Close to threshold a VCSEL operates on the fundamental transverse mode, while for larger injection the dynamics is often multimode, with the optical feedback inducing either in-phase or anti-phase transverse mode oscillations. In this paper, we study numerically the influence of current modulation on these different feedback-induced transverse-mode regimes. The modulation amplitude and period are taken as control parameters. We find that the in-phase and anti-phase regimes are robust under weak modulation. As the modulation amplitude increases, there is a transition to a dynamics governed by the current modulation, where the total output power follows the injection current and there is either single-mode or in-phase multimode behavior. However, the effect of the current modulation depends on the modulation period. Under fast modulation, the laser cannot follow the modulation and the optical-feedback-induced effects are dominant. On the contrary, under slow modulation there is a superposition of modulation and feedback effects, with the total output following the modulated current and an underlying transverse-mode behavior mainly determined by the optical feedback. A resonant behavior was observed for modulation periods close to the internal oscillation period. In this case, current modulation induces pulsing output intensity with single-mode or in-phase multimode behavior.