The buckling of thin soft ferromagnetic elastic plates in a uniform transverse magnetic field is discussed. Based on a linearized version of the general magnetoelastic equations, a plate equation is derived together with the pertinent boundary conditions. By searching a nontrivial solution of this homogeneous system, a buckling value can be found. As an example, the circular plate is considered. In this paper, Maxwell stresses are used, but it is shown that identical results can be obtained by using other models (especially Amperian current) for the magnetoelastic stresses.