This study is devoted to the implications of scale-dependent gravity in Cosmology. Redshift-space distortion data indicate that there is a tension between Λ CDM and available observations as far as the value of the rms density fluctuation, σ8, is concerned. It has been pointed out that this tension may be alleviated in alternative theories in which gravity is weaker at redshift z∼ 1. We study the evolution of density perturbations for non-relativistic matter on top of a spatially flat FLRW Universe, and we compute the combination A= fσ8 in the framework of scale-dependent gravity, where both Newton’s constant and the cosmological constant are allowed to vary with time. Upon comparison between available observational data (supernovae data as well as redshift-space distortion data) and theoretical predictions of the model, we determine the numerical value of σ8 that best fits the data.