We introduce the notion of FI-mono-retractable modules which is a generalization of compressible modules. We investigate the properties of such modules. It is shown that the rings over which every cyclic module is FI-monoretractable are simple Noetherian V -ring with zero socle or Artinian semisimple. The last section of the paper is devoted to the endomorphism rings of FI-retractable modules.
