We have performed a detailed study of the intrinsic electrical conduction process in individual monolayers of chemically reduced graphene oxide down to a temperature of 2 K. The observed conductance can be consistently interpreted in the framework of two-dimensional variable-range hopping in parallel with electric-field-driven tunneling. The latter mechanism is found to dominate the electrical transport at very low temperatures and high electric fields. Our results are consistent with a model of highly conducting graphene regions interspersed with disordered regions, across which charge carrier hopping and tunneling are promoted by strong local electric fields.