The influence of different types of attractive pair-potentials in density functional models of homogeneous nucleation is analyzed. The models considered here are based on the local-density approximation of the free-energy functional and make use of the same hard-sphere reference fluid (the Carnahan-Starling approximation). The analyzed magnitude is the ratio between the obtained density functional theory nucleation barrier height and the corresponding classical result. Some recent studies suggest that such an energy ratio follows a universal scaling relation which is independent of the form of the pair-potential. The present analysis confirms that the results are weakly dependent on the form of the interaction potential, provided that its asymptotic decay is strong enough as for those that are usually considered in practice: Yukawa, Lennard-Jones, Square-Well, etc. However, when the asymptotic decay at infinity of the attractive pair-potential is weak enough, the conjectured scaling behavior ceases to be applicable and the nucleation barrier ratio can be reduced significantly.