Haptic teleoperations play a key role in extending human capabilities to perform complex tasks remotely, employing a robotic system. The impact of haptics is far-reaching and can improve the sensory awareness and motor accuracy of the operator. In this context, a key challenge is attaining a natural, stable and safe haptic human-robot interaction. Achieving these conflicting requirements is particularly crucial for complex procedures, e.g. medical ones. To address this challenge, in this work we develop a novel haptic bilateral teleoperation system (HBTS), featuring a virtualized force feedback, based on the motion error generated by an admittance controlled robot. This approach allows decoupling the force rendering system from the control of the interaction: the rendered force is assigned with the desired dynamics, while the admittance control parameters are separately tuned to maximize interaction performance. Furthermore, recognizing the necessity to limit the forces exerted by the robot on the environment, to ensure a safe interaction, we embed a saturation strategy of the motion references provided by the haptic device to admittance control. We validate the different aspects of the proposed HBTS, through a teleoperated blackboard writing experiment, against two other architectures. The results indicate that the proposed HBTS improves the naturalness of teleoperation, as well as safety and accuracy of the interaction.