We compute the tree level cross section for gluon-gluon elastic scattering taking into account a dynamical gluon mass, and show that this mass scale is a natural regulator for this subprocess cross section. Using an eikonal approach in order to examine the relationship between this gluon-gluon scattering and the elastic pp and (p) over barp channels, we found that the dynamical gluon mass is of the same order of magnitude as the ad hoc infrared mass scale m(0) underlying eikonalized QCD-inspired models. We argue that this correspondence is not an accidental result, and that this dynamical scale indeed represents the onset of nonperturbative contributions to the elastic hadron-hadron scattering. We apply the eikonal model with a dynamical infrared mass scale to obtain predictions for sigma(tot)(pp,(p) over barp), rho(pp,(p) over barp), slope B-pp,B-(p) over barp, and differential elastic scattering cross section d sigma((p) over barp)/dt at Tevatron and CERN-LHC energies.Instituto de Física Teórica Universidade Estadual Paulista, 01405-900, São Paulo, SPInstituto de Física Gleb Wataghin Universidade Estadual de Campinas, 13083-970, SP, CampinasInstituto de Física de São Carlos Universidade de São Paulo, 13560-970, São Carlos, SPUniv Estadual Paulista, Inst Fis Teor, BR-01405900 São Paulo, Brazil