A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r-wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings K-x and K-y (where K-x and K-y are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling K-z using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.