Hamilton-Jacobi (HJ) reachability analysis is a widely used method for ensuring the safety of robotic systems. Traditional approaches compute reachable sets by numerically solving an HJ Partial Differential Equation (PDE) over a grid, which is computationally prohibitive due to the curse of dimensionality. Recent learning-based methods have sought to address this challenge by approximating reachability solutions using neural networks trained with PDE residual error. However, these approaches often suffer from unstable training dynamics and suboptimal solutions due to the weak learning signal provided by the residual loss. In this work, we propose a novel approach that leverages model predictive control (MPC) techniques to guide and accelerate the reachability learning process. Observing that HJ reachability is inherently rooted in optimal control, we utilize MPC to generate approximate reachability solutions at key collocation points, which are then used to tactically guide the neural network training by ensuring compliance with these approximations. Moreover, we iteratively refine the MPC generated solutions using the learned reachability solution, mitigating convergence to local optima. Case studies on a 2D vertical drone, a 13D quadrotor, a 7D F1Tenth car, and a 40D publisher-subscriber system demonstrate that bridging MPC with deep learning yields significant improvements in the robustness and accuracy of reachable sets, as well as corresponding safety assurances, compared to existing methods.