In this work, the behavior of the solutions of a bi-dimensional system of ordinary differential equations that models predator-prey interaction will be analyzed, obtained through a modification of the Leslie-Gower model, where the predator population considers hunting cooperation and the prey population is affected by a weak Allee effect. It is possible to find conditions for the coexistence of both species, as well as the possibility of periodic solutions, that imply oscillatory coexistence between species. We extend the properties already known for the case with a strong Allee effect and focus on the study of stability and local bifurcations.