Introduction: Needle-based neurosurgical procedures require high accuracy in catheter positioning to achieve high clinical efficacy. Significant challenges for achieving accurate targeting are (i) tissue deformation (ii) clinical obstacles along the insertion path (iii) catheter control. Objective: We propose a novel path-replanner able to generate an obstacle-free and curvature bounded three-dimensional (3D) path at each time step during insertion, accounting for a constrained target pose and intraoperative anatomical deformation. Additionally, our solution is sufficiently fast to be used in a closed-loop system: needle tip tracking via electromagnetic sensors is used by the path-replanner to automatically guide the programmable bevel-tip needle (PBN) while surgical constraints on sensitive structures avoidance are met. Methods: The generated path is achieved by combining the ”Bubble Bending” method for online path deformation and a 3D extension of a convex optimisation method for path smoothing. Results: Simulation results performed on a realistic dataset show that our replanning method can guide a PBN with bounded curvature to a predefined target pose with an average targeting error of 0.65 ± 0.46 mm in position and 3.25 ± 5.23 degrees in orientation under a deformable simulated environment. The proposed algorithm was also assessed in-vitro on a brain-like gelatin phantom, achieving a target error of 1.81 ± 0.51 mm in position and 5.9 ± 1.42 degrees in orientation. Conclusion: The presented work assessed the performance of a new online steerable needle path-planner able to avoid anatomical obstacles while optimizing surgical criteria. Significance: This method is particularly suited for surgical procedures demanding high accuracy on the desired goal pose under tissue deformations and real-world inaccuracies.