magnetic field can introduce rich physics into two-dimensional (2D) HOTIs. However, up to now the theoretical description is still incomplete and the experimental realization is still lacking. Here we investigate the influence of continuously varying magnetic field on 2D Su-Schriffer-Heeger lattice, which is one of the most celebrated HOTI models, and proposed a corresponding circuit quantum electrodynamics (cQED) simulator. Our numerical calculation shows that the zero energy corner modes (ZECMs), which can serve as evidence of the high order topology of the lattice, exhibit exotic and rich dependence on the imposed magnetic field and the inhomogeneous hopping strength. Moreover, by exploiting the parametric conversion method, we can establish time- and site-resolved tunable hopping constants in the proposed cQED simulator, thus providing an ideal platform for simulating the magnetic field induced topological phase transitions in 2D HOTIs. Since the high-order topological phases of the proposed model can be characterized by the existence of the ZECMs on the lattice, we further investigate the corner site excitation of the lattice in the steady state limit. Our numerical results imply that the predicted topological phase transitions can be unambiguously identified by the steady-state photon number measurement of the corner sites and their few neighbors. Requiring only current level of technology, our scheme can be readily tested in experiment and may pave an alternative way towards the future investigation of HOTIs in the presence of magnetic field, disorder, and strong correlation.