By means of generalized-ensemble Monte Carlo simulations, we investigate the thermodynamic behavior of a flexible, elastic polymer model in the presence of an attractive nanocylinder. We systematically identify the structural phases that are formed by competing monomer-monomer and monomer-substrate interactions. The influence of the relative surface attraction strength on the structural phases in the hyperphase diagram, parameterized by cylinder radius and temperature, is discussed as well. In the limiting case of the infinitely large cylinder radius, our results coincide with previous outcomes of studies of polymer adsorption on planar substrates.