We propose and study a model for tree-grass interactions in the context of savannas which are subjected to fire pressure. Several theoretical models in the literature which have highlighted the impact of fire on tree-grass interactions did not explicitly deal with the indirect feedback of dry grass biomass onto tree dynamics through fire intensity and frequency. The novelty in our work is to consider a fairly generic modeling of fire impact on woody biomass by means of a family of increasing and bounded functions of grass biomass. The characteristic feature of this family of functions is that, it could include several forms: linear as well as non-linear ones (sigmoidal or not). Since the nonlinear shape brings more diverse results than the previous attempts using a linear function, it could be used to show that several vegetation equilibria exist with some of them showing tree-grass coexistence features.We show that the number of equilibria with both grass and trees depends on the choice of the fire impact function. We also established thresholds defining the stability domains of the equilibria and highlighted some bifurcation parameters to provide numerical simulations complying with the theoretical properties of the model.