Depositional patterns characteristic of river mouths are controlled by the sediment-laden turbulent jet exiting from rivers. Here we show that jet instability, developing at river mouths with small width-to-depth ratio, can affect these patterns. Numerical simulations indicate that sediment eddy diffusivity, which is a measure of the spreading of sediments out of the jet core, depends on the interplay between sediment settling and large coherent flow structures, which are associated with unstable jets. Our results suggest that optimal conditions for eddy diffusivity are met when the time scale of sediment settling is equal to the characteristic eddy time scale. In this condition, sediments are deposited at the farthest distance from the jet centerline, thus promoting the formation of lateral deposits. Settling time scales much greater or much smaller than the eddy time scale is associated with low eddy diffusivity, which promotes the formation of a central deposit. Such eddy-mediated sediment transport is more evident in the Zone Of Established Flow than in the Zone Of Flow Establishment, since the former is characterized by fully developed vortices. A simplified model is proposed to estimate the settling and eddy time scales and to predict the optimal condition for levee formation.