Optical non-reciprocity that allows unidirectional flow of optical field is pivoted on time reversal symmetry breaking, which originates from radiation pressure because of light–matter interaction in cavity optomechanical systems. Here, the non-reciprocal transport of optical signals across two ports via three optical modes optomechanically coupled to the mechanical excitations of two nanomechanical resonators (NMRs) is studied under the influence of strong classical drive fields and weak probe fields. It is found that there exists the conversion of reciprocal to non-reciprocal signal transmission via tuning the drive fields and perfect non-reciprocal transmission of output fields is realized when the effective cavity detuning parameters are near resonant to the NMRs' frequencies. The unidirectional non-reciprocal transport is robust to the optomechanical couplings around resonance conditions. Moreover, the loss rates of cavities play an inevitable role in the unidirectional flow of signal across the two ports. Bidirectional transmission can also be controlled by the phase changes associated with the probe and drive fields along with their relative phase. This scheme may provide a foundation for the compact non-reciprocal communication and quantum information processing, thus enabling novel devices that route photons in unconventional ways such as all-optical diodes, optical transistors, and optical switches.