In the present work, we compute the spectrum of quasinormal frequencies of four-dimensional (4D) charged black holes (BHs) in Einstein-power-Maxwell (EpM) theory. In particular, we study scalar perturbations adopting the sixth-order WKB approximation. We analyze in detail, the behavior of the spectrum depending on the charge of the BH, the quantum number of angular momentum and the overtone number. In addition, a comparison is made between the results obtained here and the results valid for charged BHs in other theories as well as for the Reissner-Nordström BH. Finally, we have provided an analytical expression for the quasinormal spectrum in the eikonal limit.