We investigate the cool gas absorption in galaxy clusters by cross-correlating Mgii absorbers detected in quasar spectra from Data Release 16 of the Sloan Digital Sky Survey (SDSS) with galaxy clusters identified in the Dark Energy Spectroscopic Instrument (DESI) survey. We find significant covering fractions ($1-5\, \%$ within r500, depending on the chosen redshift interval), ∼4 − 5 times higher than around random sightlines. While the covering fraction of cool gas in clusters decreases with increasing mass of the central galaxy, the total Mgii mass within r500 is nonetheless ∼10 times higher than for SDSS luminous red galaxies (LRGs). The Mgii covering fraction versus impact parameter is well described by a power law in the inner regions and an exponential function at larger distances. The characteristic scale of the transition between these two regimes is smaller for large equivalent width absorbers. Cross-correlating Mgii absorption with photo −z selected cluster member galaxies from DESI reveals a statistically significant connection. The median projected distance between Mgii absorbers and the nearest cluster member is ∼200 kpc, compared to ∼500 kpc in random mocks with the same galaxy density profiles. We do not find a correlation between Mgii strength and the star formation rate of the closest cluster neighbour. This suggests that cool gas in clusters, as traced by Mgii absorption, is: (i) associated with satellite galaxies, (ii) dominated by cold gas clouds in the intracluster medium, rather than by the interstellar medium of galaxies, and (iii) may originate in part from gas stripped from these cluster satellites in the past.