This study investigates the brain functional connectivity in the rest and sleep states. We collected EEG, EOG, and fNIRS signals simultaneously during rest and sleep phases. The rest phase was defined as a quiet wake-eyes open (w_o) state, while the sleep phase was separated into three states; quiet wake-eyes closed (w_c), non-rapid eye movement sleep stage 1 (N1), and non-rapid eye movement sleep stage 2 (N2) using the EEG and EOG signals. The fNIRS signals were used to calculate the cerebral hemodynamic responses (oxy-, deoxy-, and total hemoglobin). We grouped 133 fNIRS channels into five brain regions (frontal, motor, temporal, somatosensory, and visual areas). These five regions were then used to form fifteen brain networks. A network connectivity was computed by calculating the Pearson correlation coefficients of the hemodynamic responses between fNIRS channels belonging to the network. The fifteen networks were compared across the states using the connection ratio and connection strength calculated from the normalized correlation coefficients. Across all fifteen networks and three hemoglobin types, the connection ratio was high in the w_c and N1 states and low in the w_o and N2 states. In addition, the connection strength was similar between the w_c and N1 states and lower in the w_o and N2 states. Based on our experimental results, we believe that fNIRS has a high potential to be a main tool to study the brain connectivity in the rest and sleep states.