We study the CO line luminosity (L-CO), the shape of the CO spectral line energy distribution (SLED), and the value of the CO-to-H-2 conversion factor in galaxies in the Epoch of Reionization (EoR). For this aim, we construct a model that simultaneously takes into account the radiative transfer and the clumpy structure of giant molecular clouds (GMCs) where the CO lines are excited. We then use it to post-process state-of-the-art zoomed, high resolution (30 pc), cosmological simulation of a main-sequence (M-* approximate to 10(10) M-circle dot, SFR approximate to 100M(circle dot) yr(-1)) galaxy, 'Althaea', at z approximate to 6. We find that the CO emission traces the inner molecular disc (r approximate to 0.5 kpc) of Althaea with the peak of the CO surface brightness co-located with that of the [C-II] 158 mu m emission. Its LCO(1-0) = 10(4.85) L-circle dot is comparable to that observed in local galaxies with similar stellar mass. The high (Sigma(gas) approximate to 220M(circle dot) pc(-2)) gas surface density in Althaea, its large Mach number (M approximate to 30) and the warm kinetic temperature (T-k approximate to 45 K) of GMCs yield a CO SLED peaked at the CO(7-6) transition, i.e. at relatively high-J and a CO-to-H-2 conversion factor alpha(CO) approximate to 1.5M(circle dot) (K km s(-1) pc(2))(-1) lower than that of the Milky Way. The Atacama Large Millimeter/submillimeter Array observing time required to detect (resolve) at 5 sigma the CO(7-6) line from galaxies similar to Althaea is approximate to 13 h (approximate to 38 h).