The presence of asymptotic giant branch (AGB) stars in clusters provides key constraints for stellar models, as has been demonstrated with historical data from the Magellanic Clouds. In this work, we look for candidate AGB stars in M31 star clusters from the Panchromatic Hubble Andromeda Treasury survey. Our photometric criteria selects stars brighter than the tip of the red giant branch, which includes the bulk of the thermally pulsing AGB stars as well as early-AGB stars and other luminous cool giants expected in young stellar populations (e.g., massive red supergiants, and intermediate-mass red helium-burning stars). The AGB stars can be differentiated, a posteriori, using the ages already estimated for our cluster sample. 937 candidates are found within the cluster aperture radii, half (∼450) of which are very likely cluster members. Cross-matching with additional databases reveals two carbon stars and 10 secure variables among them. The field-corrected age distribution reveals the presence of young supergiants peaking at ages smaller than 108 yr, followed by a long tail of AGB stars extending up to the oldest possible ages. This long tail reveals the general decrease in the numbers of AGB stars from initial values of ∼ 50 × 10-6 M⊙-1 at 108 yr down to ∼ 5 × 10-6 M⊙-1 at 1010 yr. Theoretical models of near-solar metallicity reproduce this general trend, although with localized discrepancies over some age intervals, whose origin is not yet identified. The entire catalog is released together with finding charts to facilitate follow-up studies.