Quantum electrodynamics (QED) effects may be included in physical processes of magnetar and pulsar magnetospheres with strong magnetic fields. Involving the quantum corrections, Maxwell electrodynamics is modified to nonlinear electrodynamics. In this work, we study the force-free magnetosphere in nonlinear electrodynamics in a general framework. The pulsar equation describing a steady and axisymmetric magnetosphere is derived, which now admits solutions with corrections. We derive the first-order nonlinear corrections to the near-zone dipole magnetosphere in some popular nonlinear effective theories. The field lines of the corrected dipole tend to converge on the rotational axis so that the fields in the polar region are stronger compared to the pure dipole case.