Phase separation in a biological cell nucleus occurs in a heterogeneous environment filled with a high density of chromatins and thus it is inevitably influenced by interactions with chromatins. As a model system of nuclear body formation in a cell nucleus filled with chromatins, we simulate the phase separation of a low-density Lennard-Jones (LJ) fluid interacting with a long, condensed polymer chain. The influence of the density variation of LJ particles above and below the phase boundary and the role of attractive interactions between LJ particles and polymer segments are investigated at a fixed value of strong self-interaction between LJ particles. For a density of LJ particles above the phase boundary, phase separation occurs and a dense domain of LJ particles forms irrespective of interactions with the condensed polymer chain whereas its localization relative to the polymer chain is determined by the LJ-polymer attraction strength. Especially, in the case of moderately weak attractions, the domain forms separately from the polymer chain and subsequently associates with the polymer chain. When the density is below the phase boundary, however, the formation of a dense domain is possible only when the LJ-polymer attraction is strong enough, for which the domain grows in direct contact with the interacting polymer chain. In this work, different growth behaviors of LJ particles result from the differences in the density of LJ particles and in the LJ-polymer interaction, and this work suggests that the distinct formation of activity-dependent and activity-independent nuclear bodies (NBs) in a cell nucleus may originate from the differences in the concentrations of body-specific NB components and in their interaction with chromatins.