A five-dimensional coupled states (5D-CS) approximation is used to compute cross sections and rate coefficients for CO+H2 collisions. The 5D-CS calculations are benchmarked against accurate six-dimensional close-coupling (6D-CC) calculations for transitions between low-lying rovibrational states. Good agreement between the two formulations is found for collision energies greater than 10cm−1. The 5D-CS approximation is then used to compute two separate databases which include highly excited states of CO that are beyond the practical limitations of the 6D-CC method. The first database assumes an internally frozen H2 molecule and allows rovibrational transitions for v ≤ 5 and j ≤ 30, where v and j are the vibrational and rotational quantum numbers of the initial state of the CO molecule. The second database allows H2 rotational transitions for initial CO states with v ≤ 5 and j ≤ 10. The two databases are in good agreement with each other for transitions that are common to both basis sets. Together they provide data for astrophysical models which were previously unavailable.