  We study the perturbative stability of magnetic black holes in a general
class of nonlinear electrodynamics, where the Lagrangian is given by a general
function of the field strength of electromagnetic field $F_{\mu\nu}$ and its
Hodge dual $\widetilde{F}_{\mu\nu}$. We derive sufficient conditions for the
stability of the black holes. We apply the stability conditions to Bardeen's
regular black holes, black holes in Euler-Heisenberg theory, and black holes in
Born-Infeld theory. As a result, we obtain a sufficient condition for the
stability of Bardeen's black holes, which restricts
$F_{\mu\nu}\widetilde{F}^{\mu\nu}$ dependence of the Lagrangian. We also show
that black holes in Euler-Heisenberg theory are stable for a sufficiently small
magnetic charge. Moreover, we prove the stability of black holes in the
Born-Infeld electrodynamics even when including
$F_{\mu\nu}\widetilde{F}^{\mu\nu}$ dependence.
