Cool gas (T∼10 4 K) traced by hydrogen Ly𝛼 emission is now routinely detected around 𝑧 ∼ 3 quasars, but little is known about their molecular gas reservoirs. Here, we present an APEX spectroscopic survey of the CO(6-5), CO(7-6) and [C ](2-1) emission lines for 9 quasars from the QSO MUSEUM survey which have similar UV luminosities, but very diverse Ly𝛼 nebulae. These observations ( rms = 2.6 mJy in 300 km s −1 ) detected three CO(6-5) lines with 3.4≤ 𝐼 CO(6−5) ≤5.1 Jy km s −1 , 620≤FWHM≤707 km s −1 , and three [C ](2-1) lines with 2.3≤ 𝐼 [C ] (2−1) ≤15.7 Jy km s −1 , 329≤FWHM≤943 km s −1 . For the CO and [C ] detected sources, we constrain the molecular gas reservoirs to be M H 2 = (0.4−6.9) ×10 11 M , while the non-detections imply M H 2 < 1.1 × 10 11 M . We compare our observations with the extended Ly𝛼 properties to understand the link between the cool and the molecular gas phases. We find large velocity shifts between the bulk of Ly𝛼 and the molecular gas systemic redshift in five sources (from ∼-400 to ∼ +1200 km s −1 ). The sources with the largest shifts have the largest Ly𝛼 line widths in the sample, suggesting more turbulent gas conditions and/or large-scale inflows/outflows around these quasars. We also find that the brightest (𝐼 [C ] (2−1) = 15.7 ± 3.7 Jy km s −1 ) and the widest (FWHM∼900 km s −1 ) lines are detected for the smallest and dimmest Ly𝛼 nebulae. From this, we speculate that host galaxy obscuration can play an important role in reducing the ionizing and Ly𝛼 photons able to escape to halo scales, and/or that these systems are hosted by more massive halos.