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With a view toward ferromagnetic-resonance-mediated detection of magnetic nanoparticles, we use micromagnetic simulations to study how the stray magnetic field produced by a uniformly magnetized spherical magnetic nanoparticle can modify the vortex gyrotropic resonance. We also show how these modifications depend on the particle's position and size. For small particle sizes, a core confinement effect induced by interaction between the particle's highly localized out-of-plane stray field and the vortex core magnetization can induce large frequency shifts. However, the generation of large shifts via this mechanism relies on the core's trajectory being localized directly below the particle. For larger particles (which generate stronger but less localized stray fields), changes in the gyrotropic frequency result from a combination of wide-scale out-of-plane spin canting and in-plane-field-induced shifting of the vortex core within its anharmonic confining potential. Compared to confinement-driven frequency shifting, the generation of large frequency shifts (10 MHz) via these latter mechanisms is less dependent on particle positioning and the core orbit radius. The use of large particles, thus, represents a more reliable sensing modality if the particles cannot be guaranteed to align centrally over the disk.