We report a new lattice density functional theory for polymer solutions at the solid-liquid interface. The theory accounts for the nearest-neighbor interactions and the long-range correlations due to chain connectivity. A Helmholtz free-energy functional is developed with an exact free-energy functional expression for the ideal chains and a thermodynamic model of lattice polymer solutions for the excess contributions. The local and weighted density approximations are used to calculate the contributions due to the athermal entropy of mixing and the internal energy of mixing, respectively. Mayer function and propagator formalism are adopted to obtain the segment-density distributions for various conformations including adsorbed trains, tails, loops, and free polymers. The predicted density distributions of polymer adsorption are in good agreement with simulation results. The results imply that as a counterbalance between energy and conformational entropy, the weighted density approximation used in the functional can rationally capture the segment-segment correlations.