The analogue of the hydrophobic hydration is explored for Lennard-Jones solutions. The free energy of solvation and its temperature derivatives, both in the constant-pressure process and in the constant-volume process, are obtained numerically for a variety of the size and energy parameters for the solute-solvent Lennard-Jones potential. We identify in the parameter space a region in which the solvation is of hydrophobic character, with an understanding that hydrophobicity is characterized by both the solvation free energy being positive and the solvation process being exothermic. Such a region is found in each case of the isobaric and isochoric conditions and the region is seen to be much wider in the isochoric process than in the isobaric one. Its origin and implication are discussed.