Possible phases in a nematic liquid crystal confined to a spherical submicrometer droplet embedded in a solid polymer are analyzed in terms of a Landau-de Gennes theory. For a droplet with a radial structure we show that the strength of the nematic-polymer interfacial interaction affects the nematic-paranematic (partially ordered isotropic phase) phase transition and may in addition induce a boundary-layer nematic phase. This boundary layer phase exists only in a narrow (approximately 0.1 K) temperature interval above the nematic phase for a restricted range of interfacial interactions. Also in the radial structure the degree of ordering is suppressed close to the center of the droplet where a defect is located. As the size of the droplet decreases, the relative size of this region of suppressed ordering increases. Below a critical radius R(c) (0.22-mu-m for 4-n-pentyl-4'-cyanobiphenyl), if the surface interaction is above a critical value (q(max) = 1.85 x 10(-3)), the transition between the nematic phase and the paranematic phase no longer occurs. A three-dimensional phase diagram is presented to demonstrate the effect of the surface interaction strength, droplet radius, and sample temperature on the stability of phases within a droplet.