The Laboratoire National des Champs Magnétiques Intenses (LNCMI) is a member of the European Magnetic Field Laboratory (EMFL). This is the French facility enabling researchers to perform experiments in the highest possible magnetic fields. The numerical modeling jointly developed with the Center of Modeling and Simulation of Strasbourg (Cemosis) plays an essential role in the understanding and the optimization of such magnets. In this context we have implemented a full 3D multi-physics model coupling thermoelectric, magnetostatic and mechanic dedicated to the study of LNCMI resistive high field magnets. We have proposed a parallel algorithm to efficiently compute the magnetic field in the zone of interest for researchers using Biot and Savart law. Validated by dedicated campaigns of measurements, this model has confirmed the 3D behaviour of the magnetic field which has been observed in some experiments such as magnetic levitation. In order to design, control or perform advanced analysis - e.g. uncertainty quantification, sensitivity analysis - many queries or real-time evaluations of a our 3D finite element models are required. We propose in this paper a reduced basis methodology applied to the Biot-Savart law in order to reduce by several order of magnitudes the computational cost of the 3D finite element model. Not only does it provide an invaluable tool for magnet designers but it also provides direct access to non-expert in numerical simulation for example to visualize the 3D magnetic field maps in real time while changing the current density. The description of this tool will be illustrated by numerical results on real magnet geometries.