In microwave breast cancer imaging magnetic nanoparticles have been recently proposed as contrast agent. Due to the non-magnetic nature of human tissues, magnetic nanoparticles make possible the overcoming of some limitations of conventional microwave imaging techniques, thus providing reliable and specific diagnosis of breast cancer. In this paper, a Compressive Sensing inspired inversion technique is introduced for the reconstruction of the magnetic contrast induced within the tumor. The applicability of Compressive Sensing theory is guaranteed by the fact that the underlying inverse scattering problem is linear and the searched magnetic perturbation is sparse. From the numerical analysis, performed in realistic conditions in 3D geometry, it has been pointed out that the adoption of this new tool allows improving resolution and accuracy of the reconstructions, as well as reducing the number of required measurements.