Adaptive cruise control (ACC) systems are designed to provide longitudinal assistance to enhance safety and driving comfort by adjusting vehicle velocity to maintain a safe distance between the host vehicle and the preceding vehicle. Generally, using model predictive control (MPC) in ACC systems provides high responsiveness and lower discomfort by solving real-time constrained optimization problems but results in computational load. This paper presents an architecture of deep learning based on model predictive control in ACC systems to avoid real-time optimization problems required by MPC, which in turn, reduces computational load. The learning dataset is acquired from the simulation data of the input/output of the MPC controller. We designed the proposed deep learning controller using long short-term memory networks (LSTMs) and simulated it in MATLAB/Simulink using the vehicle's characteristics from the advanced vehicle simulator (ADVISOR). Finally, the safety and driving comfort are compared with the PID-based control to demonstrate the performance of the proposed deep-learning architecture.