X-ray-emitting, close binary systems are among the first objects discovered in the X-ray sky and now understood to be ubiquitous in the Universe. As such, X-ray binaries can serve as a useful tool to study the evolution of their parent stellar populations, on scales from star clusters to galaxy clusters. We report observational evidence for the presence of intra-cluster X-ray sources that are not associated with the main stellar content of the individual galaxies nor with the cosmic X-ray background in the two nearest galaxy clusters, Virgo and Fornax, based primarily on archival Chandra observations. We discuss the origin of these sources, in terms of supernova-kicked low-mass X-ray binaries (LMXBs), LMXBs in globular clusters, LMXBs associated with the diffuse intra-cluster light, tidally-stripped nucleated dwarf galaxies and free-floating massive black holes. The discovery of intra-cluster X-ray sources opens a new avenue for studying the structural growth in galaxy clusters. With the large sky area of Virgo and Fornax, the all-sky survey by the eROSITA mission is expected to find at least 10 times more intracluster X-ray sources, which will form a statistically meaningful sample for exploring the physical properties and formation history of the ICL in the two clusters.