This paper presents a suggestion-based fuel efficient controller for connected and automated vehicles (CAVs) in presence of human-driven vehicles (HDVs). The suggestion-based controller, apart from evaluating the fuel efficient control solution of the host CAV, provides suggested velocity commands to the HDVs so that the fuel efficiency of itself or the group can improve. We assume that in the connected vehicle system, the HDVs are also able to receive information though Vehicle to Vehicle (V2V) communication and they always try to follow the suggested commands. The suggestion-based control provides additional decision variables to the CAVs with which they can influence the actions of the HDVs and hence improve the fuel efficiency of the whole group. The controller is implemented in a model predictive control (MPC) framework where the suggested command velocities are held constant over some prescribed time so that the driver gets enough time to reach the suggested command velocities. For this control method to function, we present a model that captures the response of a HDV to different suggested-commands. The parameters of the model is obtained from a table-top drive simulator. The accuracy of this model is also validated with the experimental data (table-top drive simulator) and the results are presented in this paper. Simulation studies for the control strategies show the efficacy of the proposed control strategy when compared with existing baseline methods.