Medical advancements in neonatology have significantly increased the number of high-risk preterm survivors. However, recent long-term follow-up studies have suggested that preterm infants are at risk for behavioral, educational, and emotional problems. Although clear relationships have been demonstrated between preterm infants and developmental problems during childhood and adolescence, less is known about the early indications of these problems. Recently, numerous studies on resting-state functional connectivity (RSFC) have demonstrated temporal correlations of activity between spatially remote cortical regions not only in healthy adults but also in neuropathological disorders and early childhood development. In order to compare RSFC of the cerebral cortex between preterm infants at term-equivalent ages and full-term neonates without any anatomical abnormality risk during natural sleep, we used an optical topography system, which is a recently developed extension of near-infrared spectroscopy. We clarified the presence of RSFC in both preterm infants and full-term neonates and showed differences between these groups. The principal differences were that on comparison of RSFC between the bilateral temporal regions, and bilateral parietal regions, RSFC was enhanced in preterm infants compared with full-term neonates; whereas on comparison of RSFC between the left temporal and left parietal regions, RSFC was enhanced in full-term neonates compared with preterm infants. We also demonstrated a difference between the groups in developmental changes of RSFC related to postmenstrual age. Most importantly, these findings suggested that preterm infants and full-term neonates follow different developmental trajectories during the perinatal period because of differences in perinatal experiences and physiological and structural development.