Diffusion of adsorbates on transition metal nanoparticles is a precursor process for heterogeneously catalyzed reactions, and as a result, an atomistic understanding of the diffusion mechanism is very important. We systematically studied adsorption and diffusion of atomic and diatomic species H, C, N, O, CO, and NO on nanometer-sized Pt and Cu nanoparticles with different sizes and shapes using density functional theory calculations. We show that nanoparticles bind adsorbates more strongly than the corresponding extended single crystal metal surfaces. We find that there is a Bronsted-Evans-Polanyi type linear correlation between the transition-state energy and the initial-state energy for adsorbate diffusing across the edges of Pt and Cu nanoparticles. We further show that the barrier for adsorbate diffusion across the nanoparticles edges can be estimated by the binding energy of the adsorbate on the nanoparticles. These results provide useful insights for understanding diffusion mediated chemical reactions catalyzed by transition metal nanoparticles which are widely used inheterogeneous catalysis.