The nature of the first massive stars may be inferred by investigating the origin of the extremely metal-poor (EMP) stars, likely formed from the ejecta of one or a few previous massive stars. We investigate the rotational properties of early massive stars by comparing the abundance patterns of EMP stars with rotating massive stellar models. Low metallicity 20 $M_{odot}$ stellar models with initial rotation rates between 0 and $70~%$ of the critical velocity are computed. Explosions with strong fallback are assumed. The ejected material is considered to fit individually the abundance patterns of 272 EMP stars with $-4<$ [Fe/H] $<-3$. With increasing initial rotation, the [C/H], [N/H], [O/H], [Na/H], [Mg/H] and [Al/H] ratios in the massive star ejecta are gradually increased. Among the 272 EMP stars considered, $sim 40-50~%$ are consistent with our models. About $60 - 70~%$ of the CEMP star sample is reproduced against $sim 20 - 30~%$ for the C-normal EMP star sample. The CEMP stars are preferentially reproduced with a material coming from mid to fast rotating massive stars. The velocity distribution derived from the best massive star models increases from no rotation to fast rotation. The maximum is reached for massive stars having initial equatorial velocities of $sim 550 - 640$ km~s$^{-1}$. Although subject to significant uncertainties, these results suggest that the rotational mixing operating in between the H-burning shell and the He-burning core of early massive stars played an important role in the early chemical enrichment of the Universe. The comparison of the velocity distribution derived from the best massive star models with velocity distributions of nearby OB stars suggests a greater amount of massive fast rotators in the early Universe. This may have important consequences for reionization or integrated light from high redshift galaxies.
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