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. We derive the muon densities as functions of energy at reference distances of 600 m and 800 m for primary energies between 2.5 PeV and 40 PeV and between 9 PeV and 120 PeV, respectively, at an atmospheric depth of about $690\,\mathrm{g/cm}^2$. The measurements are consistent with the predicted muon densities obtained from Sibyll 2.1 assuming any physically reasonable cosmic ray flux model. However, comparison to the post-LHC models QGSJet-II.04 and EPOS-LHC shows that the post-LHC models yield a higher muon density than predicted by Sibyll 2.1 and are in tension with the experimental data for air shower energies between 2.5 PeV and 120 PeV.