Microwave-induced thermoacoustic imaging (MTI) has the advantages of high resolution, high contrast, non-ionization, and non-invasive. Recently, MTI was used in the field of breast cancer screening. In this paper, based on the finite element method (FEM) and COMSOL Multiphysics software, a three-dimensional breast cancer model suitable for exploring the MTI process is proposed to investigate the influence of Young’s modulus (YM) of breast cancer tissue on MTI. It is found that the process of electromagnetic heating and initial pressure generation of the entire breast tissue is earlier in time than the thermal expansion process. Besides, compared with normal breast tissue, tumor tissue has a greater temperature rise, displacement, and pressure rise. In particular, YM of the tumor is related to the speed of thermal expansion. In particular, the larger the YM of the tumor is, the higher the heating and contraction frequency is, and the greater the maximum pressure is. Different Young’s moduli correspond to different thermoacoustic signal spectra. In MTI, this study can be used to judge different degrees of breast cancer based on elastic imaging. In addition, this study is helpful in exploring the possibility of microwave-induced thermoacoustic elastic imaging (MTAE).