We present a measurement of the density of GeV muons in near-vertical air showers using three years of data recorded by the IceTop array at the South Pole. Depending on the shower size, the muon densities have been measured at lateral distances between 200 and 1000 m. From these lateral distributions, we derive the muon densities as functions of energy at reference distances of 600 and 800 m for primary energies between 2.5 and 40 PeV and between 9 and 120 PeV, respectively. The muon densities are determined using, as a baseline, the hadronic interaction model Sibyll 2.1 together with various composition models. The measurements are consistent with the predicted muon densities within these baseline interaction and composition models. The measured muon densities have also been compared to simulations using the post-LHC models EPOS-LHC and QGSJet-II.04. The result of this comparison is that the post-LHC models together with any given composition model yield higher muon densities than observed. This is in contrast to the observations above 1 EeV where all model simulations yield for any mass composition lower muon densities than the measured ones. The post-LHC models in general feature higher muon densities so that the agreement with experimental data at the highest energies is improved but the muon densities are not correct in the energy range between 2.5 and about 100 PeV.