Predictive planning is a key capability for robots to efficiently and safely navigate populated environments. Particularly in densely crowded scenes, with uncertain human motion predictions, predictive path planning, and control can become expensive to compute in real time due to the curse of dimensionality. With the goal of achieving pro-active and legible robot motion in shared environments, in this paper we present HuMAN-MPC, a computationally efficient algorithm for Human Motion Aware Navigation using fast embedded Model Predictive Control. The approach consists of a novel model predictive control (MPC) formulation that leverages a fast state-of-the-art optimization backend based on a sequential quadratic programming real-time iteration scheme while also providing feasibility monitoring. Our experiments, in simulation and on a fully integrated ROS-based platform, show that the approach achieves great scalability with fast computation times without penalizing path quality and efficiency of the resulting avoidance behavior.