In this work the problem of vibrations and stability of rectangular plate made of magnetostrictive material in an external constant longitudinal magnetic field is studied. It is assumed that properties of the area surrounding the body is identical to the vacuum properties. On the basis of corresponding equations and conditions of the nonlinear theory of magnetoelasticity and using the linearization procedure the system of linear equations and boundary conditions describing the behavior of small disturbances in the magnetically active deformable medium are obtained. From the obtained system the characteristics of the unperturbed state are found. After, the obtained three-dimensional equations of magnetoelastic stability is brought to the two-dimensional one using the hypothesis on non-deformable normals. To solve this two-dimensional problem, asymptotic integration method is applied. Based on the solutions of the finally obtained boundary-value problem, the effect of magnetoelasticity is investigated.