An analysis based upon the Rouse bead-spring model and the Smoluchowski many particle diffusion equation in the free draining limit is utilized to derive an expression for the steady-state permeation rate of homopolymeric chainlike molecules passing through a narrow pore with a cross-sectional area slightly greater than that of an individual chain segment in an otherwise impenetrable barrier that separates in general two different solvent environments. The analysis is also applied to determine the steady-state permeation rate of homopolymers diffusing across a planar liquid-liquid interface formed by two immiscible solvents. The planar interface corresponds to a pore whose cross-sectional area is much larger than the mean square end to end dimensions of the polymer molecule. The key results give the permeation rate, J, as a function of the degree of polymerization, N, the individual bead diffusion coefficient(s) D-i (i = 1, 2) in the different solvents and a localized pore or interfacial surface contact resistance, h. The polymer-pore permeation rate becomes J similar or equal to D-R/N-3 where D-R = D1D2/(D-1 + D-2) whenever h/D-R much greater than 1 and is consistent with the prediction of reptation dynamics.