We address cluster size effects on the lambda temperature (T-lambda) for the rounded-off transition for the Bose-Einstein condensation and for the onset of superfluidity in (He-4)(N) clusters of radius R-0=aN(1/3), where a=3.5 Angstrom is the constituent radius. The phenomenological Ginsburg-Pitaevskii-Sobaynin theory for the order parameter of the second-order phase transition, in conjunction with the free-surface boundary condition, results in a scaling law for the cluster size dependence of T-lambda, which is defined by the maximum of the specific heat and/or from the onset of the finite fraction of the superfluid density. This size scaling law (T-lambda(0)-T-lambda)/T(lambda)(0)proportional toR(0)(-1/nu)proportional toN(-1/3nu), where nu (=0.67) is the critical exponent for the superfluid fraction and for the correlation length for superfluidity in the infinite bulk system, implies the depression of the finite system T-lambda relative to the bulk value of T-lambda(0). The quantum path integral molecular dynamics simulations of Sindzingre, Ceperley, and Klein [Phys. Rev. Lett. 63, 1601 (1989)] for N=64, 128, together with experimental data for specific heat of He-4 in porous gold and in other confined systems [J. Yoon and M. H. W. Chan, Phys. Rev. Lett. 78, 4801 (1997); G. M. Zahssenhaus and J. D. Reppy, ibid. 83, 4800 (1999)], are accounted for in terms of the cluster size scaling theory (T-lambda(0)-T-lambda)/T-lambda(0)=(pixi(0)/a)N-3/2(-1/2), where xi(0)=1.7+/-0.3 Angstrom is the "critical" amplitude for the correlation length in the bulk. The phenomenological theory relates T-lambda for the finite system to the correlation length xi(T) for superfluidity in the infinite bulk system, with the shift (T-lambda(0)-T-lambda) being determined by the ratio R-0/xi(T), in accord with the theory of finite-size scaling. (C) 2003 American Institute of Physics.