We present a physically motivated model to estimate the molecular hydrogen (H2) content of high-redshift (z≈ 5.7 and 6.6) Lyman α emitters (LAEs) extracted from a suite of cosmological simulations. We find that the H2 mass fraction, , depends on three main LAE physical properties: (a) star formation rate, (b) dust mass and (c) cold neutral gas mass. At z≈ 5.7, the value of peaks and ranges between 0.5 and 0.9 for intermediate-mass LAEs with stellar mass M*≈ 109 - 1010 M⊙, decreasing for both smaller and larger galaxies. However, the largest value of the H2 mass is found in the most luminous LAEs. These trends also hold at z≈ 6.6, although, due to a lower dust content, when averaged over all LAEs; they arise due to the interplay between the H2 formation/shielding controlled by dust and the intensity of the ultraviolet Lyman–Werner photodissociating radiation produced by stars. We then predict the carbon monoxide (CO) luminosities for such LAEs and check that they are consistent with the upper limits found by Wagg et al. for two z > 6 LAEs. At z≈ 5.7 and 6.6, the lowest CO rotational transition observable for both samples with the actual capabilities of the Atacama Large Millimeter Array (ALMA) is the CO(6−5). We find that at z≈ 5.7, about 1–2 per cent of LAEs, i.e. those with an observed Lyman α luminosity larger than 1043.2 erg s−1, would be detectable with an integration time of 5–10 h (a signal-to-noise ratio of 5); at z≈ 6.6, none of the LAEs would be detectable in CO, even with an ALMA integration time of 10 h. We also build the CO ‘flux function’, i.e. the number density of LAEs as a function of the line-integrated CO flux, SCO, and show that it peaks at SCO= 0.1 mJy at z= 5.7, progressively shifting to lower values at higher redshifts. We end by discussing the model uncertainties.