The ARCADE2 and LWA1 experiments have claimed an excess over the cosmic microwave background (CMB) at low radio frequencies. If the cosmological high-redshift contribution to this radio background is between 0.1 per cent and 22 per cent of the CMB at 1.42 GHz, it could explain the tentative EDGES low-band detection of the anomalously deep absorption in the 21-cm signal of neutral hydrogen. We use the upper limit on the 21-cm signal from the Epoch of Reionization (z = 9.1) based on 141 h of observations with LOFAR to evaluate the contribution of the high-redshift Universe to the detected radio background. Marginalizing over astrophysical properties of star-forming haloes, we find (at 95 per cent CL) that the cosmological radio background can be at most 9.6 per cent of the CMB at 1.42 GHz. This limit rules out strong contribution of the high-redshift Universe to the ARCADE2 and LWA1 measurements. Even though LOFAR places limit on the extra radio background, excess of 0.1-9.6 per cent over the CMB (at 1.42 GHz) is still allowed and could explain the EDGES low-band detection. We also constrain the thermal and ionization state of the gas at z = 9.1, and put limits on the properties of the first star-forming objects. We find that, in agreement with the limits from EDGES high-band data, LOFAR data constrain scenarios with inefficient X-ray sources, and cases where the Universe was ionized by stars in massive haloes only.