Context. The majority of the Wolf-Rayet (WR) stars represent the stripped cores of evolved massive stars who lost most of their hydrogen envelope. Wind stripping in single stars is expected to be inefficient in producing WR stars in metal-poor environments such as the Small Magellanic Cloud (SMC). While binary interaction can also produce WR stars at low metallicity, it is puzzling that the fraction of WR binaries appears to be about 40%, independent of the metallicity. Aim. We aim to use the recently determined physical properties of the twelve known SMC WR stars to explore their possible formation channels through comparisons with stellar models. Methods. We used the MESA stellar evolution code to construct two grids of stellar models with SMC metallicity. One of these consists of models of rapidly rotating single stars, which evolve in part or completely chemically homogeneously. In a second grid, we analyzed core helium burning stellar models assuming constant hydrogen and helium gradients in their envelopes. Results. We find that chemically homogeneous evolution is not able to account for the majority of the WR stars in the SMC. However, in particular the apparently single WR star SMC AB12, and the double WR system SMC AB5 (HD 5980) appear consistent with this channel. We further find a dichotomy in the envelope hydrogen gradients required to explain the observed temperatures of the SMC WR stars. Shallow gradients are found for the WR stars with O star companions, while much steeper hydrogen gradients are required to understand the group of hot apparently single WR stars. Conclusions. The derived shallow hydrogen gradients in the WR component of the WR+O star binaries are consistent with predictions from binary models where mass transfer occurs early, in agreement with their binary properties. Since the hydrogen profiles in evolutionary models of massive stars become steeper with time after the main sequence, we conclude that most of the hot (Teff>60kK ) apparently single WR stars lost their envelope after a phase of strong expansion, e.g., as the result of common envelope evolution with a lower mass companion. The so far undetected companions, either main sequence stars or compact objects, are then expected to still be present. A corresponding search might identify the first immediate double black hole binary progenitor with masses as high as those detected in GW150914.