Abstract We present our new Atacama Large Millimeter/submillimeter Array (ALMA) observations targeting CO(6–5) emission from three luminous Lyman-break galaxies (LBGs) at z spec = 6.0293–6.2037 found in the Subaru/Hyper Suprime-Cam survey, whose O iii 88 μ m and C ii 158 μ m emissions have been detected with ALMA. We find a marginal detection of the CO(6–5) line from one of our LBGs, J0235–0532, at the ≃4 σ significance level and obtain upper limits for the other two LBGs, J1211–0118 and J0217–0208. Our z = 6 luminous LBGs are consistent with the previously found correlation between the CO luminosity and the infrared luminosity. The unique ensemble of the multiple far-infrared emission lines and underlying continuum fed to a photodissociation region model reveals that J0235–0532 has a relatively high density of hydrogen nuclei n H that is comparable to those of low- z (U)LIRGs, quasars, and Galactic star-forming regions with high n H values, while the other two LBGs have lower n H consistent with local star-forming galaxies. By carefully taking account of various uncertainties, we obtain constraints on total gas mass and gas surface density from their CO luminosity measurements. We find that J0235–0532 is located below the Kennicutt–Schmidt (KS) relation, comparable to the z = 5.7 LBG, HZ10, previously detected with CO(2–1). Combined with previous results for dusty starbursts at similar redshifts, the KS relation at z = 5–6 is on average consistent with the local one.