A nonclassical theory of nucleation, based on the density-functional (DF) approach, is developed for the gas-liquid transitions of two-dimensional (2D) Lennard-Jones (LJ) fluids. The methods of Weeks-Chandler-Andersen perturbation theory are used to approximate the LJ potential with a temperature-dependent hard-disk diameter plus an attractive tail. The resulting free energy functional is then used to calculate the free energy barrier to nucleation. We find that the curvature of the 2D nucleus is not important to the rate of nucleation (in contrast to the 3D counterpart). The effect of curvature is readily inferred from the ratio of nucleation rate from classical Becker-Doring theory to that from DF theory. Our calculation suggests that classical nucleation theory. actually works reasonably well for 2D LJ fluids in predicting the temperature-dependence of the nucleation rate (whereas for 3D LJ fluids it fails badly).