Recently, researchers have shown an increased interest in considering plants as a model of inspiration for designing new robot locomotions. Growing robots, that imitate the biological growth presented by plants, have proved irresistible in unpredictable and distal environments due to their morphological adaptation and tip-extension capabilities. However, as a result of the irreversible growing process exhibited by growing robots, classical control schemes could fail in obtaining feasible solutions that respect the permanent growth constraint. Thus, in this article, a Nonlinear Model Predictive Control (NMPC) scheme is proposed to guarantee the robot's performance towards point stabilization while respecting the constraints imposed by the growing process and the control limits. The proposed NMPC-based growth control has applied to the kinematic model of the recently proposed plant-inspired robots in the literature, namely, vine-like growing robots. Numerical simulations have been performed to show the effectiveness of the proposed NMPC-based growth control in terms of point stabilization, disturbance rejection, and obstacle avoidance and encouraging results were obtained. Finally, the robustness of the proposed NMPC-based growth control is analyzed against various input disturbances using Monte-Carlo simulations that could guide the tuning process of the NMPC.