Thermoplastic magnetorheological elastomers (MRE) are soft magnetic materials with multi stimuli response. As the name suggests, thermoplastic MREs are formed by the suspension of magnetic particles in a thermoplastic elastomer matrix, whose rheological properties can be controlled by temperature. The composite itself can thus be controlled by temperature and magnetic field. In this article, we present the method and results of quasistatic and dynamic mechanical properties of thermoplastic polyurethane MREs (TPU-MREs) based on the observations made on the influence of hardener and plasticizer content from the composition point of view and the directional interdependence of strain and magnetic field on the magnetorheological (MR) effect. Quasistatic tensile and torsion tests followed by dynamic oscillation tests conducted within the linear elastic regime at three different temperatures with and without externally applied magnetic field are performed on the TPU-MRE samples. The tests are conducted with an inhouse developed test setup adapted to a commercial rheometer. The obtained results demonstrate a strong correlation to the change in moduli of the TPU-MRE with respect to change in composition and temperature. From the results obtained, one can conclude that samples with lower hardener content with plasticizer have the highest relative MR effect.