We extend the mean-field Gaussian chain theory, originally developed for non-dilute solutions of athermal polymer chains in a slit, to solutions in a channel with a square cross section. The formulation allows one to calculate the monomer density profile, the chemical potential of the confined polymer chain, and therefore the partition coefficient. For the mean-field potential, we used the first-order approximation that neglects local monomer density fluctuations and the second-order approximation that takes into account the fluctuations. The results of the density profile and the partition coefficient were compared with those obtained in the lattice Monte Carlo simulations. The theoretical results obtained with the first-order approximation agreed well with the simulation results for chains of 100 beads below the average monomer density of ca. 0.2. At higher concentrations, the second-order results gave a better agreement. This cross over indicates a change in the interactions between polymer chains from those in one-dimension to those in three-dimensions as the correlation length in the confined solution becomes sufficiently shorter than the channel width.