This article presents a model predictive controller (MPC) developed for the autonomous steering of concentric tube robots (CTRs). State-of-the-art CTR control relies on differential kinematics developed by local linearization of the CTRs mechanics model and cannot explicitly handle constraints on robot’s joint limits or unstable configurations commonly known as snapping points. The proposed nonlinear MPC explicitly considers constraints on the robot configuration space (i.e., joint limits) and the robot’s workspace (i.e., mixed boundary conditions on robot curvature). Additionally, the MPC calculates control decisions by optimizing the model-based predictions of future robot configurations. This way, it avoids configurations it cannot recover from, i.e., joint limits, singular configurations, and snapping. The proposed controller is evaluated via simulations and experimental studies with a variety of trajectories of increasing complexity. Simulation results demonstrate the capability of MPC to avoid singularities while satisfying robot mechanical constraints. Experimental results demonstrate that our solution enables following of trajectories unattainable by state-of-the-art controllers with mean error corresponding to 1% of robot arclength.