The theoretical model of proteins on the two-dimensional square lattice, introduced previously, is extended to include the hydrophobic interactions. Two proteins, whose native conformations have different folded patterns, are studied. Units in the protein chains are classified into polar units and nonpolar units. If there is a vacant lattice point next to a nonpolar unit, it is interpreted as being occupied by solvent water and the entropy of the system is assumed to decrease by a certain amount. Besides these hydrophobic free energies, the specific long-range interactions studied in previous papers are assumed to be operative in a protein chain. Equilibrium properties of the folding and unfolding transitions of the two proteins are found to be similar, even though one of them was predicted, based on the one globule model of the transitions, to unfold through a significant intermediate state (or at least to show a tendency toward such a behavior), when the hydrophobic interactions are strongly weighted. The failure of this prediction led to the development of a more refined model of transitions; a non-interacting local structure model. The hydrophobic interactions assumed here have a character of non-specific long-range interactions. Because of this character the hydrophobic interactions have the effect of decelerating the folding kinetics. The deceleration effect is less pronounced in one of the two proteins, whose native conformation is stabilized by many pairs of medium-range interactions. It is therefore inferred that the medium-range interactions have the power to cope with the decelerating effect of the non-specific hydrophobic interactions.