We study the structure around a dilute Lennard-Jones (LJ) solute in a supercritical LJ fluid using integral equation theories and Monte Carlo simulation. We show that the Percus-Yevick approximation can be seriously in error in the near-critical region. The solute's coordination number as a function of the solvent density can be nearly linear, or quite non-linear, depending on the solute-solvent interaction. These results are rationalized as a crossover from a low-density energy-dominated regime to a high-density entropy-(packing-) dominated regime, and are discussed within the context of the local density enhancement concept.