The presence of spectral curvature resulting from the synchrotron self-absorption of extragalactic radio sources could break down the spectral smoothness feature. This leads to the premise that the bright radio foreground can be successfully removed in 21-cm experiments that search for the epoch of reionization (EoR). We present a quantitative estimate of the effect of the spectral curvature resulting from the synchrotron self-absorption of extragalactic radio sources on the measurement of the angular power spectrum of the low-frequency sky. We incorporate a phenomenological model, which is characterized by the fraction (f) of radio sources with turnover frequencies in the range of 100–1000 MHz and by a broken power law for the spectral transition around the turnover frequencies νm, into simulated radio sources over a small sky area of 10° × 10°. We compare statistically the changes in their residual maps with and without the inclusion of the synchrotron self-absorption of extragalactic radio sources after the bright sources of mJy are excised. Furthermore, the best-fitting polynomials in the frequency domain on each pixel are subtracted. It has been shown that the effect of synchrotron self-absorption on the detection of the EoR depends sensitively on the spectral profiles of the radio sources around the turnover frequencies νm. A hard transition model, described by the broken power law with the turnover of spectral index at νm, would leave pronounced imprints on the residual background and would therefore cause serious confusion with the cosmic EoR signal. However, the spectral signatures on the angular power spectrum of the extragalactic foreground, generated by a soft transition model in which the rising and falling power laws of the spectral distribution around νm are connected through a smooth transition spanning ≥200 MHz in a characteristic width, can be fitted and consequently subtracted by the use of polynomials to an acceptable degree (δT < 1 mK). As this latter scenario seems to be favoured in both theoretical expectations and radio spectral observations, we conclude that the contamination of extragalactic radio sources by synchrotron self-absorption in 21-cm experiments is probably very minor.