We report Monte Carlo simulation results for the thermodynamic and structural properties of binary nonadditive mixtures of hard spheres with not too dissimilar sizes. The simulations were carried out over a range of parameters where fluid-fluid separation is known to compete with freezing. We investigated the onset of structural ordering in a one-phase scheme as that provided by the residual multiparticle entropy, an integrated measure of the weight of spatial correlations involving more than two particles in the configurational entropy of the system. This approach has been already tested in the symmetric case [Saija, F.; Pastore, G.; Giaquinta, P. V. J. Phys. Chem. B 1998, 102, 10368]. The present study confirms the indication that a rather small amount of nonadditivity may significantly influence the phase behavior of the mixture also in a thermodynamic regime where osmotic depletion effects do not play a leading role.