Using a large scale model of macaque V1, we have shown (Wielaard & Sajda 2003, 2005) how only local short-range (<0.5 mm) connections within V1 can mediate surround suppression and account for a large fraction of the magnitude of suppression seen experimentally. In our model surround suppression arises from one of three mechanisms: (A) an increase in cortical inhibition, (B) a decrease in cortical excitation, or (C) both of these simultaneously. It is known that LGN neurons exhibit both classical and extraclassical surround suppression, with classical surround suppression observed at lower spatial frequencies. This leaves open the question of whether the unexplained fraction of suppression seen in our model could be inherited from the LGN or instead requires considering long-range lateral connections and/or extrastriate feedback. Using our model, we consider the effect of classical LGN surround suppression on suppression in V1 cortical neurons, in particular by measuring the distribution of the suppression index at spatial frequencies that are a quarter of those typically used to optimally drive cortical neurons. We find that at these lower spatial frequencies, nearly all of the classical surround suppression of LGN cells is transferred to V1 cells, either via direct LGN input into the cell and/or suppression of recurrent cortico-cortical excitation. We also see that at these low spatial frequencies, the prevalence of the cortical mechanisms for surround suppression is shifted in favor of mechanisms B and C, which rely on the reduction of excitation. This shift occurs at the expense of mechanism A, which relies on direct inhibition. Our model thus predicts 1) a substantial increase in V1 surround suppression is possible by sufficiently lowering the stimulus spatial frequency and 2) that ultimately the magnitude of surround suppression seen in V1 neurons is explainable by considering only the short-range cortical connections and the LGN input.