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). To 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 stateof-the-art zoomed, high resolution (30 pc), cosmological simulation of a main-sequence (M * ≈ 10 10 M , SF R ≈ 100 M yr −1 ) galaxy, "Althaea", at z ≈ 6. We find that the CO emission traces the inner molecular disk (r ≈ 0.5 kpc) of Althaea with the peak of the CO surface brightness co-located with that of the [4.85 L is comparable to that observed in local galaxies with similar stellar mass. The high (Σ gas ≈ 220 M pc −2 ) gas surface density in Althaea, its large Mach number (M≈ 30), and the warm kinetic temperature (T k ≈ 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 α CO ≈ 1.5 M (K km s −1 pc 2 ) −1 lower than that of the Milky Way. The ALMA observing time required to detect (resolve) at 5σ the CO(7-6) line from galaxies similar to Althaea is ≈ 13 h (≈ 38 h).