One of the challenges relating to acoustic metamaterials is to achieve a tunable performance without modifying the structure. In this paper, we propose two types of acoustic metamaterials with a magnetorheological elastomer (MRE), and their tunable band gap structures and the transmission spectra are investigated by the finite element method (FEM). The MRE acts as a cladding layer, and its shear modulus can be changed by an externally applied magnetic field. The cell resonance frequency of acoustic metamaterial is changed. The band gap structures and the transmission spectra of the two kinds of acoustic metamaterials are calculated under various magnetic fields, and it is found that the frequency and width of band gap, the maximum attenuation frequency and transmission loss of transmission spectrum increase with externally applied magnetic field intensity increasing. Meanwhile, two types of the mass-spring models are used to estimate the band gap frequencies of the two kinds of acoustic metamaterials. The FEM results are in good agreement with the estimation results. In addition, the effects of material parameters of core and shell and filling rate on the band gap and transmission spectrum are also studied. The effects of core material parameters on the band gap and transmission spectrum of single-layer acoustic metamaterial are analyzed. It is found that the core mass has an effect on the band gap frequency and width, and the elastic parameter of the core affects the transmission loss of the transmission spectrum. The influences of core and shell material parameter on the band gap and transmission spectrum of double-layer acoustic metamaterial is analyzed by the control variable method. The results show that the core and shell mass affect the band gap frequency, width and pass-band width, and the elastic parameter of the core and the shell affect the transmission loss of the transmission spectrum. As the filling rate increases, the band gap frequency and width of the single- and double-layer MRE acoustic metamaterial increase, the maximum attenuation frequency of the transmission spectrum does not change, and the transmission loss increases. These results will greatly contribute to the application of acoustic metamaterials to controlling the active noise and vibration.