The purpose of this study is to consider an approximate method for a two-dimensional magnetostatic boundary value inverse problem, in which two kinds of boundary conditions with errors are simultaneously imposed on a part of the boundary of a bounded domain. The problem is recast to a variational problem to identify the proper boundary condition for the rest of the boundary. Minimization of an objective functional with a regularization term using the steepest descent method leads our variational problem to an iterative process. Based on numerical computations using the edge finite elements, it is concluded that an estimated solution for the boundary data without errors is in good agreement with an exact one, and the regularization term yields good approximate solutions for the boundary data with errors.