This paper is devoted to investigating the existence of traveling wavefront solutions with a sufficiently large wave speed for a nonlocal diffusion model with nonlocal delay effect using a perturbation method. Our proof is based on an abstract formulation of the wave profile as a solution to an operator equation in a specific Banach space, combined with the Fredholm theory and the Banach contraction mapping principle. Several numerical simulations are presented to illustrate our main results.