The Enskog theory for the self-diffusion coefficient for fluids with continuous potentials, such as the Lennard-Jones, is developed. Starting from the Green-Kubo formula (rather than the conventional kinetic equation) and introducing the similar assumptions upon which the Boltzmann equation is based, we derived a general expression for the memory kernel and the self-diffusion coefficient. The numerical analysis is implemented for the Lennard-Jones fluid. The time-dependent memory kernel is calculated and compared with the latest molecular dynamics simulations. Excellent agreement is obtained at the low density. The self-diffusion coefficient is evaluated for various temperatures and densities. The ratio of the Enskog self-diffusion coefficient to the simulation value is plotted against density. Significant difference of this density dependence from that for the hard-sphere fluid is observed. In particular, the well-known maximum observed (in the diffusion versus density plot) for the hard sphere fluid is found to be completely absent in the Lennard-Jones fluid. Our results reduce to the conventional Chapman-Enskog expression in the low density limit and can be applicable to the systems with singular potentials such as the hard sphere.