In this work, our aim is to investigate the star formation rate (SFR) of the host galaxies of active galactic nuclei (AGNs) in the early Universe. To this end, we constructed a sample of 149 luminous (L2 − 10 keV > 1044 erg s−1) X-ray AGNs at z ≥ 3.5 selected in three fields with different depths and observed areas from the Chandra COSMOS Legacy survey, XMM-XXL North, and eROSITA Final Equatorial-Depth Survey. We built their spectral energy distributions (SED) using the available multi-wavelength photometry from X-rays up to the far-IR. Then, we estimated the stellar mass, M*, and the SFR of the AGNs using the X-CIGALE SED fitting algorithm. After applying several quality criteria, we ended up with 89 high-z sources. More than half (55%) of the X-ray sample have spectroscopic redshifts. Based on our analysis, our high-z X-ray AGNs live in galaxies with a median M* = 5.6 × 1010 M⊙ and SFR* ≈ 240 M⊙yr−1. The majority of the high-z sources (∼89%) were found inside or above the main sequence (MS) of star-forming galaxies. Estimations of the normalised SFR, SFRNORM, defined as the ratio of the SFR of AGNs, to the SFR of MS galaxies, show that the SFR of AGNs is enhanced by a factor of ∼1.8 compared to non-AGN star-forming systems. Combining our results with previous studies at lower redshifts, we confirmed that SFRNORM does not evolve with redshift. Using the specific black hole accretion rate (i.e. LX divided by M*), λBHAR, which can be used as a tracer of the Eddington ratio, we found that the majority of AGNs that lie inside or above the MS have higher specific accretion rates compared to sources below the MS. Finally, we found indications that the SFR of the most massive AGN host galaxies (log (M*/M⊙) > 1011.5 − 12) remains roughly constant as a function of M*, in agreement with the SFR of MS star-forming galaxies.