This paper studies the magneto-thermal problems with the nonlinear material law. The main difficulty is to analysis the Joule heating term γ(u)| ∇×H|2. First, a regularized model is introduced. By time discretization, the well-posedness of the discrete problem is established, and the convergence of the solution as the time step size τ→0 is deduced. Finally, the solution to the regularized problem converges to the original as ε→0. The mathematical analysis of this paper provide a routine to obtain the well-posedness of the magneto-thermal problems and gives an answer to the open question from the previous work.