Abstract The role of the environment on the formation of S0 galaxies is still not well understood, specifically in the outskirts of galaxy clusters. We study eight low-redshift clusters, analyzing galaxy members up to cluster-centric distances of ∼2.5 R 200. We perform 2D photometric bulge–disk decomposition in the g, r, and i bands from which we identify 469 double-component galaxies. We separately analyze the colors of the bulges and disks and their dependence on the projected cluster-centric distance and local galaxy density. For our sample of cluster S0 galaxies, we find that bulges are redder than their surrounding disks, show a significant color–magnitude trend, and have colors that do not correlate with environment metrics. On the other hand, the disks associated with our cluster S0s become significantly bluer with increasing cluster-centric radius but show no evidence for a color–magnitude relation. The disk color–radius relation is mainly driven by galaxies in the cluster core at 0 ≤ R/R 200 < 0.5. No significant difference is found for the disk colors of backsplash and infalling galaxies in the projected phase space (PPS). Beyond R 200, the disk colors do not change with the local galaxy density, indicating that the colors of double-component galaxies are not affected by preprocessing. A significant color–density relation is observed for single-component disk-dominated galaxies beyond R 200. We conclude that the formation of cluster S0 galaxies is primarily driven by cluster core processes acting on the disks, while evidence of preprocessing is found for single-component disk-dominated galaxies. We publicly release the data from the bulge–disk decomposition.