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ABSTRACT We present results from simultaneous modelling of 2D (projected along the line of sight) position, proper motion, and line-of-sight velocity for Gaia- and Apache Point Observatory Galactic Evolution Experiment (APOGEE)-observed stars near the centre of the Sagittarius (Sgr) dwarf spheroidal galaxy. We use a mixture model that allows for independent sub-populations contributed by the Sgr galaxy, its nuclear star cluster Messier 54 (M54), and the Milky Way foreground. We find an offset of $$0.295\pm 0.029$$ deg between the inferred centroids of Sgr and M54, corresponding to a (projected) physical separation of $$0.135\pm 0.013$$ kpc. The detected offset might plausibly be driven by unmodelled asymmetry in Sgr’s stellar configuration; however, standard criteria for model selection favour our symmetric model over an alternative that allows for bilateral asymmetry. We infer an offset between the proper motion centres of Sgr and M54 of $$[\Delta \mu _{\alpha }\cos \delta ,\Delta \mu _{\delta }]=[4.9, -19.7] \pm [6.8, 6.2] \ \mu \mathrm{ as}\,\mathrm{ yr}^{-1}$$ ($$[0.61, -2.46] \pm [0.85, 0.77] \ \mathrm{ km}\,\mathrm{ s}^{-1}$$), with magnitude similar to the covariance expected due to spatially correlated systematic error. We infer an offset of $$4.1\pm 1.2 \ \mathrm{ km}\,\mathrm{ s}^{-1}$$ in line-of-sight velocity. Using inferred values for the systemic positions and motions of Sgr and M54 as initial conditions, we calculate the recent orbital history of a simplified Sgr/M54 system, which we demonstrate to be sensitive to any line-of-sight distance offset between M54 and Sgr, and to the distribution of dark matter within Sgr.