In a nonlinear teleoperation system controlled for task-space position tracking, while the time-varying delay in the communication channel has been addressed, the actuator saturation has not been taken into account yet. Considering that in practice, the actuator saturation is a serious constraint, disregarding it in the controller design stage can cause problems. In this paper, we have proposed a control framework to ensure end-effectors position tracking while satisfying sub-task control in the presence of the nonlinear dynamics for the telemanipulators, bounded time-varying delays in the communication channels and saturation in the actuators. We have shown that in free motion and when the operator applies a bounded force to the local robot, the proposed controller not only guarantees the position convergence of the end-effectors but also guarantees the accomplishment of the sub-task control. The efficiency of the proposed control algorithm is validated showing a number of numerical simulations.